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Valuation Rings in Ore Extensions
Journal of Algebra 235, 665᎐680 Ž2001. doi:10.1006rjabr.2000.8484, available online at http:rrwww.idealibrary.com on
Valuation Rings in Ore Extensions H. H. Brungs1 Uni¨ ersity of Alberta, Edmonton, Alberta T6G 2G1, Canada
and M. Schroder ¨ Uni¨ ersitat ¨ Duisburg, Lotharstrasse 65, D-47048, Duisburg, Germany Communicated by Susan Montgomery Received June 18, 1999
Extensions of valuation rings V of a skew field K are considered in the skew field F s K Ž x, . for a monomorphism of K. At least two such extensions exist if is an automorphism, but no extension may exist if is a monomorphism only. There exist extensions R Ž a. of V in F with xRŽ a. s aRŽ a. for every non-zero a in K if and only if V is invariant and is compatible with V. 䊚 2001 Academic Press
1. INTRODUCTION MacLane described in wMx the set of real-valued extensions of a Žrealvalued. valuation of a ground field F0 in a simple transcendental extension F1 s F0 Ž x .. There always exist infinitely many such extensions and some related results can be found in wRx and wBOx; see also wBx and wOx. Many other authors have considered the problem of how to extend a valuation on a field F0 to a simple transcendental extension F1 , for example, Matignon and Ohm wMO1, MO2x, Alexandru et al. wAPZx, and Khanduja wKx. Haase used the fact that a complete discrete valuation of the center F of a finite-dimensional division algebra D can be extended to D in order to compute the Brauer group of local fields. In contrast to the commutative situation, not every valuation of the center can be extended to a finite-dimensional division algebra; see for example wBGx. 1
The first author is supported by NSERC. 665 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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We say that the pair Ž K, V . is a valued skew field if K is a skew field with the subring V such that a g K _V implies ay1 g V; we also say that V is a valuation ring of K. Let be a monomorphism of K and F s K Ž x, . the corresponding Ore extension of K. We consider the set of extensions of V in F, i.e., the valuation rings R of F with R l K s V. The results are surprisingly different from the commutative situation. Even though at least two extensions of V exist in F if is an automorphism, see Theorem 2.2, an example is given, see 4.2, where is a monomorphism and V has no extension in F. In Section 3 the important condition that is compatible with V is discussed and V has extensions in F in this case. It is proved in 4.3 that an invariant rank-one valuation ring V has exactly two extensions in F s K Ž x, . for a non-compatible automorphism. In the special case V s K, an automorphism, all extensions of V are determined in 4.6. In Section 5 it is shown that for all a g K _ 04 there exist extensions R of V in F with xR s aR if and only if is compatible and V is invariant; see Theorem 5.1. Ultra products of extensions of a valuation ring V are considered in 5.3.
2. STANDARD EXTENSIONS Let K be a skew field with subring V such that a g K _V implies ay1 g V. We then say that the pair Ž K, V . is a valued skew field and that V is a valuation ring of K. We use UŽ V . to denote the group of units of V, and J Ž V . s V _UŽ V . is the maximal ideal of V. A valued skew field Ž F, R . is an extension of Ž K, V . if K : F is a skew field extension and R l K s V; we also say in this case that R is an extension of V in F. Let be a monomorphism of K and K w x, x s Ý x ia i ¬ a i g K 4 be the skew polynomial ring with ax s x Ž a. defining the multiplication where a g K. This ring is a right Ore domain; see wC, p. 54x, and hence F s K Ž x, . s K w x, x Sy1 , the skew field of quotients of K w x, x, exists, where S s K w x, x_ 04 . For V a valuation ring of K we investigate the set of extensions R of V in F, with F a simple Ore extension of K. The next well-known result shows that can be extended to an automorphism ˜ of a sub skew field K : K˜ of F such that F s K˜Ž x, ˜ .. 2.1. LEMMA. Gi¨ en a skew field K with monomorphism of K, then K˜ s D ⬁ns 0 x n Kxyn is a sub skew field of F s K Ž x, . and can be extended to an automorphism ˜ of K˜ such that F s K˜Ž x, ˜ .. Proof. If Tn denotes the skew field x n Kxyn , then Tn : Tnq1 , since x Kxyn s x n Ž x Ž K . xy1 . xyn s x nq1 Ž K . xyŽ nq1. : x nq1 KxyŽ nq1.. It follows that K˜ s jTn is a skew field contained in F. The mapping ˜ , with ˜ Ž x naxyn . s x n Ž a. xyn for a g K, is well defined and an automorphism n
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˜ Clearly, K˜ : F, and for Ýkis0 x ia˜i s 0, a˜i g K, ˜ there exists an n with of K. i . n n yn yn Ž a s x a x for a g K and all i. Hence, x Ý x a ˜i ˜i x s Ý x iai s 0 shows i i that a i s 0 for i s 0, . . . , k. It follows that F s K Ž x, . s K˜Ž x, ˜ .. If V is a valuation ring of K and if there exists an extension R of V in ˜ This suggests that conversely we F, then R l K˜ is an extension of V in K. might try to determine first the extensions V˜ of V in K˜ and then the extensions of the various V˜ in F s K˜Ž x, ˜ . where in this last step we know that ˜ is an automorphism. The next result shows that in the case where is an automorphism of K there always exist at least two extensions of V in F. In order to prove this theorem we consider skew power series rings and Laurent series rings. Let K be a skew field with automorphism . Then K ww x, xx s Ý⬁is0 x ia i ¬ a i g K 4 is the ring of skew power series which contains the right and left Ore set S s x i ¬ i G 04 . The multiplication in K ww x, xx is determined by ax s x Ž a. for all a in K. The localization K Ž Ž x, . . s Sy1 K w x, x s
½
⬁
Ý iGyn
x iai ai g K
5
is the skew field of skew Laurent series. For a non-zero element ␣ g K ŽŽ x, .., ␣ s Ý⬁iGyn x ia i , we define ¨ Ž ␣ . s k s min i ¬ a i / 04 and w Ž ␣ . s a k , the lowest non-zero coefficient of ␣ . 2.2. THEOREM.
Let be an automorphism of the skew field K and V a
¨ aluation ring of K. Then V has at least the two standard extensions R Ž1. and R Žy1. in F s K Ž x, .: The ¨ aluation ring R Ž1. of F is characterized as the
extension of V in F with the property that xa g J Ž R Ž1. . for all a g K, or equi¨ alently by the condition Ži.
xRŽ1. ; aRŽ1. for all 0 / a g K.
The ¨ aluation ring R Žy1. of F is characterized as the extension of V in F with the property that xy1 a g J Ž R Žy1. . for all a in K, or equi¨ alently by the condition Žii.
aRŽy1. ; xR Žy1. for all a g K.
Proof. Consider Fˆ s K ŽŽ x, .., the skew field of skew Laurent series which contains the skew field F s K Ž x, .. The subring Rˆ s V q ˆ It follows from the definition that xK ww x, xx is a valuation ring of F: ␣ g Fˆ is in Rˆ if ¨ Ž ␣ . ) 0 or if ¨ Ž ␣ . s 0 and w Ž ␣ . g V. If ¨ Ž ␣ . s 0 and Ž . w Ž ␣ . s a0 g K _V, then ay1 0 g V and from ␣ s a 0 1 y m , m g xK ww x, xx, it follows that ␣y1 s Ž1 q m q m2 q ⭈⭈⭈ . ay1 is an element in 0 ˆ Finally, if ¨ Ž ␣ . - 0, then ¨ Ž ␣y1 . ) 0 and ␣y1 g R. ˆ This shows that R.
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ˆ Rˆ. is a valued skew field. Let R Ž1. [ F l Rˆ and R Ž1. l K s Rˆ l K s V Ž F, follows; i.e., R Ž1. is an extension of V in F. Clearly, condition Ži. holds for R Ž1. since ay1 xR Ž1. ; R Ž1. for all 0 / a g K, and this condition, by the bijectivity of , is equivalent to the property that xa g J Ž R . for all 0 / a g K. To prove that property Ži. characterizes R Ž1. , let R be any extension of V in F with xa g J Ž R . and hence x na g J Ž R . for all n G 1 and a g K. It follows that all expressions of the form Ž1 q xc1 q ⭈⭈⭈ qx n c n . with c i g K are units in R. Every element ␣ in F can be written as ␣ s Ž a0 q xa1 q ⭈⭈⭈ qx na n . Ž b 0 q xb1 q ⭈⭈⭈ qx m bm . X
X
y1
X X. s x k a Ž 1 q xaX1 q ⭈⭈⭈ qx n aXnX . Ž 1 q xbX1 q ⭈⭈⭈ qx m bm
y1
for a i , aXi , bi , bXi , a g K and k g ⺪. We use the fact that is an automorphism to prove the second equation. It follows that ␣ is an element of R if and only if k ) 0 or k s 0 and a g V. This proves the uniqueness of R and hence R s R Ž1. . On the other hand, we can consider FˆŽy1. s K ŽŽ y, .. for y s xy1 and s y1 , which is a skew field that contains F. It also contains the valuation ring RˆŽy1. s V q yK ww y, xx, and R Žy1. [ RˆŽy1. l F is an extension of V in F with property Žii.. The proofs of the statements about R Žy1. are similar to the proofs given above for R Ž1. . 2.3. Remark. With the assumptions as in Theorem 2.2 let 0 / ␣ g F with ␣ s Ž x na0 q x nq1 a1 q ⭈⭈⭈ qx nqk a k .Ž x m b 0 q ⭈⭈⭈ qx mqt bt .y1 where a i , bj g K and a0 / 0 / b 0 . Then ␣ g R Ž1. if and only if either n ) m or . g V. n s m and ym Ž a0 by1 0 2.4. Remark. With the assumptions as in Theorem 2.2 let RX [ V q xK w x, x : F. Then RX is a subring of K w x, x and SX [ 1 q xa1 q x 2 a2 q ⭈⭈⭈ qx na n ¬ a1 , a2 , . . . , a n g K 4 is a right Ore system in RX with R Ž1. s RX SXy1. 2.5. COROLLARY.
Let be a monomorphism of the skew field K and V a X
˜ then V has at least two ¨ aluation ring of K. If V has an extension V˜ in K, distinct extensions
RXŽ1.
and
RXŽy1.
in F which are also extensions of V˜X .
3. COMPATIBILITY We consider in this section a condition on the monomorphism of the skew field K and the valuation ring V of K which guarantees that V has an extension in F s K Ž x, .. This condition was used in wMax and wBTx to
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construct valuation rings R that have no two-sided ideals / Ž0., J Ž R . and R, i.e., where R is nearly simple. 3.1. DEFINITION. The monomorphism of k is said to be compatible with the valued skew field Ž K, V . or we also say that is compatible with V if Ž V . : V and Ž J Ž V .. : J Ž V .. 3.2. LEMMA. The following conditions are equi¨ alent: Ži. Žii. Žiii. Živ.
is compatible with Ž K, V .; for all a in K, Ž a. is in V if and only if a is in V; for all a in K, Ž a. is in UŽ V . if and only if a is in UŽ V .; for all a in K, Ž a. is in J Ž V . if and only if a is in J Ž V ..
Proof. Ži. « Žii.. That a g V implies Ž a. g V follows directly from 3.1. If, conversely, Ž a. g V for a f V, then ay1 g J Ž V . and Ž ay1 . g J Ž V . follows by Ži.; this is a contradiction, since Ž a., Ž a.y1 g UŽ V .. Žii. « Žiii. If a g UŽ V ., with b its inverse, then Ž a. g UŽ V . with Ž b . as its inverse. Conversely, if Ž a. g UŽ V ., then there exists b g UŽ V . with Ž a. b s b Ž a. s 1. By Žii. we have a g V. If a g J Ž V ., then ay1 f V and b s Ž ay1 . f V, a contradiction. Žiii. « Živ. If a g J Ž V ., then 1 y a g UŽ V . and by Žiii. it follows that 1 y Ž a. g UŽ V . and hence that Ž a. g V. If Ž a. g UŽ V ., then a g UŽ V ., a contradiction. If, conversely, Ž a. g J Ž V ., then 1 y Ž a. s Ž1 y a. g UŽ V . and 1 y a g UŽ V . follows. If a g UŽ V ., then Ž a. g UŽ V ., a contradiction. Živ. ª Ži. If a g V, Ž a. f V, then Ž ay1 . g J Ž V .. Hence ay1 g Ž . J V , a contradiction. If a g J Ž V ., then Ž a. g J Ž V . by Živ.. 3.3. Remark. The compatibility of with Ž K, V . is also equivalent to the following condition: Žv. aV ª Ž a.V, for a g K, defines a strictly monotone mapping of the totally ordered lattice of cyclic right V-submodules of K into itself. 3.4. LEMMA. Let K˜ and ˜ be as in Lemma 2.1. If is compatible with Ž K, V ., then V˜ [ D nG 0 x n Vxyn is a ¨ aluation ring of K˜ that extends V to K˜ ˜ and ˜ is compatible with V. Proof. Since x n Vxyn s x nq1 Ž V . xyŽ nq1. : x nq1 VxyŽ nq1., and since for ˜ ␣ s x naxyn with a g K for some n, it follows that either ␣ or ␣y1 ␣ g K, ˜ Hence, V˜ is a valuation ring of K. ˜ If is contained in x n Vxyn : V. ␣ s x naxyn for a g V is in V˜ l K, then n Ž ␣ . s a g V and ␣ g V ˜ To follows, which shows V˜ l K s V; i.e., V˜ is an extension of V in K. ˜ show that ˜ is V-compatible we use the condition Žii. of 3.2 and the fact
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that an element ␣ g K˜ is in V˜ if and only if there exists an n G 0 with ˜ n Ž ␣ . g V. Hence, ␣ g V˜ implies ˜ n Ž ␣ . g V for some n and ˜ n Ž ˜ Ž ␣ .. ˜ then s ˜ Ž ˜ n Ž ␣ .. g V; ˜ Ž ␣ . g V˜ follows. Conversely, if ˜ Ž ␣ . g V, ˜ n Ž ˜ Ž ␣ .. s ˜ nq1 Ž ␣ . g V; ␣ g V˜ follows.
˜ see Lemma 2.1, is 3.5. Remarks. Ži. The automorphism ˜ of K, X ˜ ˜ compatible with a valuation ring V of K that extends V if and only if is ˜ compatible with V and V˜X s V. Žii. There exist examples for compatible with V and an extension V˜X of V in K˜ which is different from V˜ and is therefore not compatible with ˜ . Proof of Ži.. Assume that ˜ is compatible with the extension V˜X of V ˜ Let a g V : V˜X , and hence ˜ Ž a. s Ž a. g V˜X l K s V. Conin K. versely, if a g K and Ž a. g V, then ˜ Ž a. g V˜X , a g V˜X l K s V which shows that is compatible with V. Since ˜ is compatible with V˜ ŽLemma ˜ For ␣ g K˜ we have ␣ s x naxyn for 3.4., it remains to show that V˜X s V. some a g K, and some n G 0. This implies ˜ n Ž ␣ . s a and ␣ g V˜X if and only if a g V, using the compatibility of ˜ with V˜X . However, this condition is equivalent to the condition that requires ␣ to be an element ˜ hence V˜X s V˜ follows. of V, To prove Žii. we consider the following example. Let K [ Ž⺡Ž x 0 , x 1 , . . . . be the function field over the rationals in countably many indeterminates with the monomorphism defined by Ž x n . s x nq1 for n s 0, 1, 2, . . . . It follows that K˜ s ⺡Ž . . . , xy2 , xy1 , x 0 , x 1 , x 2 , . . . . with ˜ Ž x k . s x kq1 for k g ⺪. For K we consider the valuation ¨ which is defined on ⺡w x 0 , x 1 , x 2 , . . . x by ¨
Ž Ý rn , n , . . . Ž x 0n 0
1
0
x 1n1 . . . . . s min
½ Ýn
i
rn 0 , n1 , . . . / 0 ;
5
let V be the corresponding valuation ring. Then is compatible with V and V has infinitely many extensions in K˜; however, ˜ is compatible only with V˜ s DnG 0 x n Vxyn. It follows from the next result that an extension R of V exists in F that contains both x and xy1 if and only if is compatible with V. 3.6. THEOREM. Let be a monomorphism and V a ¨ aluation ring of the skew field K. There exists an extension R of V in F s K Ž x, . with x, xy1 g R if and only if is compatible with Ž K, V .. Proof. Assume that R is an extension of V in F with x g UŽ R .. For a g K we have ax s x Ž a. and hence a is a unit in V if and only if Ž a. s xy1 ax is a unit in R and therefore in V. It follows from Lemma 3.2Žiii. that is compatible with V.
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Conversely, if is compatible with V it follows from Lemma 3.4 that ˜ there exists an extension V˜ of V in K˜ such that ˜ is compatible with V. i ˜ ˜ ˜ 4 The set S s Ý x a i ¬ a i g V, Ýa i V s V is a right᎐left Ore system in the skew polynomial ring V˜w x, ˜ x such that R s V˜w x, ˜ x Sy1 is a valuation ring of F Žsee wBSchx.. It follows from the construction that x is in UŽ R .. To show that R is an extension of V we consider an element ␣ in the intersection R l K which certainly contains V. We have ␣ s ŽÝ x ia i .ŽÝ x i c i .y1 for Ý x i c i g S and Ý x ia i g V˜w x, ˜ x. It follows that Ý x ia i s ␣ Ý x i c i s Ý x i i Ž ␣ . c i and at least one of the coefficients c i , say c i 0 , is in UŽ V˜ . since Ý x i c i is in S. Then i 0 Ž ␣ . s a i 0 cy1 i 0 is an element in V˜ l K s V and ␣ g V follows by using Lemma 3.2Žii.. This shows that R l K s V. 4. COMPLETE SETS OF EXTENSIONS We discuss in this section cases where all extensions of Ž K, V . in F s K Ž x, . can be determined. After collecting conditions on and Ž K, V . that guarantee the existence of an extension R of V in F, we give an example of Ž K, V . and a monomorphism of K so that V has no extension in F. We also describe a case where the extensions R Ž1. and R Žy1. that occur in Theorem 2.2 comprise exactly the set of all extensions of V in F. The extensions of V in F for V s K and an automorphism are determined in Theorem 4.6. They are R Žy1. , F, and the f-adic valuation rings R f of F for an irreducible invariant polynomial f in K w x, x; here, R Ž1. s R x . In particular, V s K has exactly three extensions if is an automorphism of K so that no power n, n G 1, is an inner automorphism. 4.1. LEMMA. There exists an extension of V in K˜ and hence in F in any of the following cases: Ža. K is commutati¨ e; Žb. is an automorphism; Žc. is compatible with Ž K, V .. Proof. It follows from Theorem 2.2 that every extension of V in K˜ can be extended to an extension of V in F. So it remains to show that V can be extended to K˜ in all three cases Ža., Žb., and Žc.. If K is commutative, ˜ If is an automorphism, then then so is K˜ and V can be extended to K. ˜ Ž . K s K. If is compatible with K, V then V can be extended to K˜ by Lemma 3.4. The result in the case Žc. also follows from Theorem 3.6. We now construct an example of a valued skew field Ž K, V . and a monomorphism of K so that V has no extension in F s K Ž x, ..
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4.2. EXAMPLE. Let ⺡Ž t . be a simple transcendental extension of the rationals and the automorphism of ⺡Ž t . defined by Ž t . s ty1 . We take K [ ⺡Ž t .Ž y, . and show that a monomorphism of K is defined by Ž y . s y q yy1 and Ž t . s t Žand so Ž a. s a for all a g ⺡Ž t ... Let ␣ s Ý y ia i be an element in ⺡Ž t .w y, x. Then Ž ␣ . s ÝŽ y q yy1 . ia i / 0 for ␣ / 0 since the non-zero term y na n will not be cancelled in Ž ␣ . if y na n is the highest non-zero term in ␣ . It follows that induces a ⺡Ž t .-linear mapping from the ⺡Ž t . right vector space ⺡Ž t .w y, x into K and that this is a ring monomorphism since y1 s and aŽ y q yy1 . s ay q ayy1 s y Ž a. q yy1y1 Ž a. s Ž y q yy1 . Ž a. for a g ⺡Ž t .. However, ⺡Ž t .w y, x is a right Ore domain and the monomorphism can be extended to the skew field K of quotients of ⺡Ž t .w y, x. Let F [ K Ž x, .. In K we consider the valuation ring V which is uniquely determined as the extension of the t-adic valuation ring ⺡w t xŽ t . with the property that ya g J Ž V . for all a g ⺡Ž t .; here we use the fact that is an automorphism and apply Theorem 2.2. We now claim that V has no extension R in F. To prove this consider the element [ xyxy1 g F. It follows that
q y1 s y y1
y1
y1 y1
Ž ).
since q s xyx q xy x s x Ž y q y x s x Ž y . x s yxxy1 y1 y1 s y. Further, ty s yt and tx s xt imply t s xyy1 Ž xy1 tx . yxy1 s y1 y1 y1 y1 y1 xy tyx s xt x s t , i.e.,
y1 t s ty1 .
y1 . y1
y1
Ž )).
If we assume that R is an extension of V in F, then y g V ; R implies that q y1 g R by Ž).. By construction we have t g V ; R and ty1 f V, hence ty1 f R. It follows from Ž)). that f UŽ R ., hence g J Ž R . or y1 g J Ž R .. However, q y1 g R, g J Ž R . lead to the contradiction y1 g R, and similarly y1 g J Ž R . implies the contradiction g R. Hence, no extension of V exists in F. We prove in the next result that under certain circumstances the two extensions R Ž1. and R Žy1. described in Theorem 2.2 are the only extensions of V in F. We say that the valuation ring V of the skew field K is archimedean if for every a in K and every j g J Ž V . there exists an n G 0 with j na, aj n g J Ž V .. 4.3. THEOREM. Assume for the ¨ alued skew field Ž K, V . that is a non-compatible automorphism of K and that V is archimedean. Then R Ž1. and R Žy1. are exactly the extensions of V in F s K Ž x, ..
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Proof. Since is an automorphism, it follows from Theorem 2.2 that R Ž1. and R Žy1. are extensions of V in F. Conversely, we have to show that any extension R of V in F is equal to either R Ž1. or R Žy1. . Since is not compatible, condition Žiii. of Lemma 3.2 does not hold and either there exists an element u g UŽ V . with Ž u. f UŽ V . or there exists an element a g K _UŽ V . with Ž a. g UŽ V .. If in the first case Ž u. f J Ž V ., then uy1 g UŽ V . and Ž uy1 . g J Ž V ., i.e., we can assume that u g UŽ V . but Ž u. g J Ž V .. We can argue in a similar way in the second case and conclude that by the non-compatibility of we have either Ži. an element u g UŽ V . with Ž u. g J Ž V .; or Žii. an element j g J Ž V . with Ž j . g UŽ V .. Now we consider first the case x g R which by the non-compatibility of and Theorem 3.6 implies that x g J Ž R ., and then the case x f R which implies xy1 g J Ž R .. In the first case we will prove that xa g J Ž R . for any a g K, which in turn implies that R s R Ž1. by Theorem 2.2. Let a g K. In the case Ži. there exists an element u g UŽ V . with Ž u. g J Ž V . and hence for any n G 0 we obtain n
xa s uyn u n xa s uyn x Ž u . a. Since V is archimedean, it follows that Ž u. na g J Ž V . for n large enough, which implies that xa g J Ž R .. In the case Žii. there exists an element j g J Ž V . with Ž j . g UŽ V . and for any n G 0 we have xa s y1 Ž a . x s y1 Ž a . j n jyn x s y1 Ž a . j n x Ž j .
yn
,
.yn
where x g R, Ž j g UŽ V .. Since V is archimedean, it follows that y1 Ž a. j n g J Ž V . for n large enough. We conclude that xa g J Ž R . for any a g K and that R s R Ž1. by Theorem 2.2. We now assume that xy1 g R. Since is an automorphism we have F s K Ž x, . s K Ž xy1 , y1 .. We apply the case considered above and obtain xy1 a g J Ž R . for all a in K, since xy1 g R and y1 is not compatible with V. Then R s R Žy1. by Theorem 2.2. In the next result we characterize archimedean valuation rings in various ways. 4.4. PROPOSITION. The following conditions are equi¨ alent for a ¨ aluation ring V / K of a skew field K : Ža. V is archimedean; Žb. V is in¨ ariant and has rank one;
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Žc. there exists a ¨ aluation ¨ from K into Ž⺢, q. j ⬁4 with V s a g K ¬ ¨ Ž a. G 04 . We recall that V is called invariant if all its one-sided ideals are two-sided or, equivalently, if dVdy1 s V for all 0 / d g K. The valuation ring V has rank n if there exist exactly n completely prime ideals / Ž0. in V. A valuation from K to Ž⺢, q. j ⬁4 is a mapping from K into this set so that ¨ Ž a. s ⬁ if and only if a s 0; ¨ Ž ab. s ¨ Ž a. q ¨ Ž b . and ¨ Ž a q b . G min ¨ Ž a., ¨ Ž b .4 for a, b in K. In the proof of the proposition we show first that Žb. implies Žc.. If V is an invariant rank-one valuation ring of K, then aV ¬ 0 / a g K 4 is a group with aVbV s abV defining its operation. This group is isomorphic to a subgroup of Ž⺢, q., and a valuation as in Žc. can be defined. The condition Žc. implies Ža. since ¨ is a real-valued valuation. To show that V has rank one if V is archimedean, we assume on the contrary that there exists a completely prime ideal P with V > J Ž V . > P > Ž0. and we choose an element 0 / p g P and an element j g J Ž V ._ P. If for some n G 0 we have j n py1 s r g J Ž R ., then j n s rp g P, hence j g P, since P is completely prime; this is a contradiction. It remains to show that an archimedean valuation ring V is invariant and it is sufficient to prove that every principal right ideal of V is two-sided. Otherwise there exists an element 0 / a in V and an element r in V with ra f aV, hence raj s a for some element j g J Ž V .. We show next that we can assume r g UŽ V .. Otherwise we have r g J Ž V . and raj s a implies Ž r y 1. aj s raj y aj s aŽ1 y j . with r y 1, 1 y j g UŽ R .. Hence, Ž r y 1. ajŽ1 y j .y1 s a or uaw s a for u s r y 1 g UŽ R . and w s jŽ1 y j .y1 g J Ž R .. Then u naw n s a implies w nay1 s ay1 uyn f J Ž V . for all n G 0; this is a contradiction. We do not know the exact conditions on Ž K, V . and the automorphism such that the two standard extensions of V are exactly the extensions of V. We give an example of a non-invariant rank-2 valuation ring V with a non-compatible automorphism so that V has more than two extensions in F. 4.5. EXAMPLE. Consider K s ⺡Ž i .ŽŽ t, .., the skew field of skew Laurent series over ⺡Ž i ., i 2 s y1, with the automorphism equal to conjugations; i.e., Ž i . s yi. Further, let V s B q t⺡Ž i .ww t, xx where B s ⺪w i xŽ2yi. is the Ž2 y i .-adic valuation ring of ⺡Ž i ., and let X be the identity mapping of K. Then F s K Ž xX , X . contains an extension R of V with xX , xXy1 g R since X is compatible with V; see Theorem 3.6. However, F s K Ž x, . also for x s txX and with Ž a. s ty1 at for a g K. It follows that is not compatible with V, since for example Ž2 q i .y1 g V,
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but ŽŽ2 q i .y1 . s Ž2 y i .y1 f V. The three extensions R, R Žy1. , and R Ž1. are distinct, since xR s tR, xR Ž1. ; tR Ž1. and xR Žy1. > tR Žy1. , where R Ž1. and R Žy1. are the standard extensions described in Theorem 2.2 for V in F s K Ž x, .. We determine all extensions R of V with V s K in F s K Ž x, . for an automorphism. All one-sided ideals in T s K w x, w are principal and a two-sided ideal I / Ž0. of T has the form I s fT s Tf for an element f s x m f 1Ž x . a, with 0 / a in K, f 1Ž x . s x na n q ⭈⭈⭈ qxa1 q 1, and f 1T s Tf 1. We say that f is an invariant element in T. Hence, f 1Ž x . b s bf 1Ž x . for all b in K and xf 1Ž x . s f 1Ž x . x. It follows that i Ž b . s a i bay1 for a i / 0 and i Ž a i . s a i for all i. If no positive power of is an inner automorphism of K then x n T, n G 0, are the only non-zero ideals of T. Otherwise, there exists r G 1 minimal with r an inner automorphism, i.e., r Ž b . s ubuy1 for all b in K and a certain element u in K. It follows that f 1Ž x . s x r k a r k q ⭈⭈⭈ qx r a r q 1 with a r i in InvŽ K . s a g K ¬ Ž a. s a4 and uyi a r i in ZŽ K ., the center of K. Conversely, every element f s x m f 1Ž x . a, with f 1Ž x . of the above form, generates an ideal fT s Tf of T ; see wJ, Chap. Ix for related results. A maximal ideal fT s Tf is completely prime only if the element f is in addition irreducible in T ; i.e., fT is also a maximal right Žor left. ideal in T. If f is an invariant and irreducible element in T then S s T _ fT is a right Žand left. Ore system in T. To see this let g g T and s be in S and gT q sT s dT, g s dg 1 , s s ds1 , and g 1 u q s1¨ s 1 for elements d, g 1 , s1 , u, ¨ in T, d, s1 g S. If g 1 u f fT, then g 1 us1 s s1Ž1 y ¨ s1 . / 0 for us1 f fT. If g 1 u g fT, then s1¨ g 1 s g 1Ž1 y ug 1 . and ug 1 g fT, 1 y ug 1 g S. The ring R f [ TSy1 exists and is an extension of V s K in F. 4.6. THEOREM. The extensions of V s K in F s K Ž x, ., an automorphism, are F, R Žy1. , and the rings R f for any irreducible in¨ ariant element f in K w x, x. Proof. Let R be any extension of V s K in F. If xy1 is in J Ž R ., then xy1 a is in J Ž R . for all a in K and R s R Žy1. by Theorem 2.2. Otherwise we have x in R and hence T s K w x, x is a subring of R. Then J Ž R . l T s N is an ideal in T and N is completely prime since J Ž R . is completely prime. Hence, either N s Ž0., J Ž R . s Ž0., and R s F or N s fT s Tf for an irreducible, invariant element f in T. If s g S s T _ N, then sy1 g R and R = R f follows. If r s hgy1 g R_ R f , we can assume that h g S and g g N. Then h g UŽ R ., g g UŽ R ., and f g UŽ R .; this is a contradiction that shows R s R f . 4.7. EXAMPLES TO ILLUSTRATE THEOREM 4.6. Ži. Assume that no power n of an automorphism of K is an inner automorphism for n G 1.
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Then K w x, x has just one irreducible invariant polynomial, f s x, and V s K has exactly three extensions in F s K Ž x, .; F, R Ž1. , and R Žy1. . One possible example is K s ⺡Ž t r ¬ r g ⺡. with Ž t r . s t 2 r defining the automorphism. Žii. In the case where K q ⺓, the complex numbers, and is the usual conjugation, it follows that 2 s id is an inner automorphism and the invariant polynomials in ⺓w x, x s T have the form f s x m f 1Ž x . a for a in ⺓ and f 1Ž x . s Ý x 2 i c 2 i for c 2 i g ⺢, c0 s 1. The monic, invariant, irreducible polynomials in T are of the form f s x or f s x 2 q r with 0 - r g ⺢. The element g s x 2 y 1 s Ž x q i .Ž x q i . generates a prime ideal of T that is not completely prime. 4.8. Remark. If RX / F is an extension of V s K in F s K Ž x, . with an automorphism, then RX s RXrJ Ž RX . is a skew field finite dimensional over K. If V is a valuation ring of K and W is an extension of V in RX , then R s a g RX ¬ a q J Ž RX . g W 4 is an extension of V in F. This construction will always produce the extensions R Ž1. and R Žy1. of V and possibly additional extensions if K w x, x contains irreducible invariant monic elements f / x. However, all extensions of V are obtained in this way only in very special cases; see also the examples in Section 5.
5. EXTENSIONS R WITH DESIGNED VALUE FOR xR Let Ž K, V . be a valued skew field, a monomorphism of K, and F s K Ž x, .. Let W Ž K . s W Ž K, V . s Ž cV ¬ 0 / c g K 4 , F. be the value set of cyclic right V-modules / Ž0. in K, totally ordered by inclusion. If R is an extension of V in F, then W Ž K . can be embedded into W Ž F, R . by identifying cV with cR for 0 / c g K, and this embedding will be order preserving. Therefore it makes sense to discuss the position of an element ␣ in F U , i.e., of ␣ R, in relation to the original value set W Ž K .; we will only consider ␣ s x. There are two cases: first, xR s cR for some c g K _ 04 , and, second, xR / cR for all c g K. We consider the first case. If K and F s K Ž x . are commutative fields and V is a valuation ring of K, then there exists for every non-zero element a in K an extension R s R Ž a. of V in F with xR s aR. Now let Ž K, V . be a valued skew field and a monomorphism of K. Then there exists for a particular non-zero element a in K an extension R s R Ž a. of V in F s K Ž x, . with xR s aR if and only if ay1 xR s R. With xX [ ay1 x and the corresponding monomorphism X defined by X Ž c . s Ž acay1 . for c in K Žsuch that cxX s xX X Ž c .., we have F s K Ž xX , X . and xX R s R. By Theorem 3.6 such an extension R of V in F
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exists if and only if X is compatible with Ž K, V .. We can now prove the following result. 5.1. THEOREM. Let Ž K, V . be a ¨ alued skew field and a monomorphism of K. Then there exists for e¨ ery non-zero element a in K an extension R s R Ž a. of V in F s K Ž x, . with xR s aR if and only if is compatible with Ž K, V . and V is in¨ ariant. Proof. If is compatible with V and V is invariant, then for every 0 / a g K we have aVay1 s V, and for any c in K it follows that c is in V if and only if X Ž c . s Ž acay1 . is in V. Hence X is compatible with V, and an extension R of V exists in F with aR s xR by the observation before the theorem. To prove the converse, we observe first that is compatible with V by Theorem 3.6, since an extension R Ž1. of V exists with xR Ž1. s R Ž1.. In addition, X with X Ž c . s Ž acay1 . is compatible with V for all 0 / a, c in K by the above remarks. This means that c is in V if and only if Ž acay1 . is in V, if and only if acay1 is in V for all 0 / a in K ; the ring V is invariant. If V is an invariant valuation ring of K, then the monomorphism of K is compatible with V, if and only if X with X Ž c . s Ž acay1 . for c g K is compatible with V for every non-zero element a in K. Hence, the next result is a consequence of Theorem 5.1 and the remarks made before this theorem. 5.2. COROLLARY. tions are equi¨ alent:
Assume that V is in¨ ariant. Then the following condi-
Ži. is compatible with V. Žii. For e¨ ery non-zero element a in K there exists an extension R s R Ž a. of V in F s K Ž x, . with xR s aR. Žiii. There exists a non-zero element a0 in K and an extension R of V in F with xR s a0 R. We consider now the second case where xR / cR for all c in K. Then xR determines a cut C s ŽU, O . in the totally ordered set W Ž K . with U [ aV g W Ž K . ¬ aR : xR4 and O [ bV g W Ž K . ¬ xR ; bR4 .
Ž ).
Here, C ŽU, O . is a cut in W Ž K . in the general sense that ŽU, O . is a partition of W Ž K . so that U j O s W Ž K . and ␣ -  for all ␣ g U and all  g O. Further, ŽU, O . s ŽU X , OX . if and only if U s U X and O s OX .
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Given any cut C s ŽU, O . in W Ž K . and an extension R of V in F, we then say that x determines C Žat R . if and only if Ž). holds. We can ask the following question: Assuming that Ž K, V . is a valued skew field, is an automorphism of K, and C s ŽU, O . is a cut in W Ž K ., does there exist an extension R of V in F s K Ž x, . so that x determines C? If U s , the empty set, then R Ž1. is an extension of V so that x determines C s Ž , W Ž K .., and similarly x determines the cut ŽW Ž K ., . at R s R Žy1. Žsee Theorem 2.2.. We can therefore assume that both U / and O / for the cut C s ŽU, O . under consideration. If U has a maximal element aV, then x determines C at an extension R s R Ž a. of V with aR s xR. This extension problem was discussed in the first part of this section; we therefore assume also that U does not have a maximal element. Then there exists a cofinal subset ␥ i s c i V ¬ i g I 4 of U with I a totally ordered index set without maximal element and ␥ i - ␥ j if and only if i - j in I. Using this notation we have the following result. 5.3. PROPOSITION. Assume that there exists for e¨ ery ␥ i s c i V an extension R i of V in F with c i R i : xR i ; bR i for all i in I and all b g K _ 04 with bV g O. Then there exists an extension R s R C of V so that aR ; xR ; bR for all a, b in K with aV g U and bV g O; i.e., x determines the cut C at R. Proof. The extension R will be obtained by using a suitable ultra product of the R i . Let F be the filter on I consisting of all subsets of I that contain a subset of the form Si 0 s i g I ¬ i G i 0 4 for some i 0 in I. Let M be an ultra filter on I containing F. In the ring Ł i g I Ž F . i s Ž a i . i g I ¬ a i g F 4 we define an equivalence relation ‘‘; ’’ with Ž a i . ; Ž bi . if and only if the set i g I ¬ a i s bi 4 is an element in M. Then F˜ s Ł i g I Ž F . ir; is a skew field with elements wŽ a i .x corresponding to the equivalence class containing the element Ž a i .. The subset ˜ since R˜ s wŽ a i .x g F˜ ¬ a i g R i for all i4 is a valuation subring of F, wŽ a i .x g F˜_ R˜ implies that wŽ a i .x / wŽ0.x and i g I ¬ a i g R i 4 s S f M. Hence, S s i g I ¬ a i f R i 4 g M and S : i ¬ 0 / a i and ay1 g R i4 g M i ˜ and wŽ a i .xy1 g R. The mapping that assigns to a in F the element wŽ a.x in F˜ is an ˜ and we can assume that F˜ is an extension of F. If embedding of F into F, wŽ a.x s wŽ ri .x is an element in F˜ l K for a in K and ri in R i , then w I g I ¬ a s ri 4 g M and a g V follows; so R s R C [ R˜ l F is an extension of V in F. It remains to prove that c i 0 R : xR ; bR for all c i 0 , i 0 g I, and all b g K with bV g O. By assumption we have extensions R i of V with c i R i :
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xR i ; bR i for all i g I and b g K with bV g O. Hence, Ž1. Ž2.
xy1 c i g R i for all i in I, and xy1 b f R i for all i in I and b g K with bV g O.
Further, for each i 0 g I and i G i 0 we have c i 0V : c i V, and so elements d i g V : R i exist with c i 0 s c i d i . By Ž1. it follows that xy1 c i 0 s xy1 c i d i g ˜ where we can R i for i G i 0 , and therefore wŽ xy1 c i 0 .x s wŽ xy1 c i d i .x g R, y1 y1 wŽ .x define d i s 0 for i - i 0 . Hence, x c i 0 s x c i 0 g R˜ l F s R, and c i 0 R : xR follows for every i 0 g I. On the other hand, we have xy1 b g F with x -1 b f R i by Ž2. for all i g I. Hence, xy1 b s wŽ xy1 b .x f R˜ and therefore xy1 b f R˜ l F s R. It follows that Ž xy1 b .y1 g J Ž R . and xR ; bR for all b g K with bV g O. Combining Theorem 5.1 and Proposition 5.3 we obtain the following result: 5.4. COROLLARY. Let Ž K, V . be a ¨ alued field and an automorphism of K. Then there exists for e¨ ery cut C s ŽU, O . of W Ž V . an extension R s R C of V so that x determines C at R if V is in¨ ariant and is compatible with V. 5.5. EXAMPLE. Let K s ⺡Ž t r ¬ r g ⺡. and be the automorphism of K with Ž t r . s t 2 r as in Example 4.7Ži.. Let V be the t-adic valuation ring of K, and V is invariant, and compatible with V. Since W Ž V . is order isomorphic to the rational numbers, it follows by Corollary 5.4 that the cardinality of the set of extensions of V in F s K Ž x, . is greater than or equal to the cardinality of the set of real numbers. On the other hand, the extensions of V obtained as in the Remark 4.8 are just the extensions R Ž1. and R Žy1. . ACKNOWLEDGMENTS We thank the referee for several helpful comments.
REFERENCES wAPZx wBOx wBx wBGx
V. Alexandru, N. Popescu, and A. Zaharescu, All valuations on K Ž x ., J. Math. Kyoto Uni¨ . 30 Ž1990., 207᎐225. N. Bourbaki, ‘‘Elements of Mathematics: Commutative Algebra,’’ Herman, Paris, 1972. R. Brown, Valuations, primes, and irreducibility in polynomial rings and rational function fields, Trans. Amer. Math. Soc. 174 Ž1972., 451᎐488. H. H. Brungs and J. Grater, Valuation rings in finite dimensional division algebras, ¨ J. Algebra 120 Ž1989., 90᎐99.
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H. H. Brungs and M. Schroder, Prime segments of skew fields, Can. J. Math. 47 ¨ Ž1995., 1148᎐1176. wBTx H. H. Brungs and G. Torner, Extensions of chain rings, Math. Z. 185 Ž1984., 93᎐104. ¨ wCx P. M. Cohn, ‘‘Free Rings and Their Relations,’’ Academic Press, London, 1985. wJx N. Jacobson, ‘‘Finite Dimensional Division Algebras over Fields,’’ Springer-Verlag, Berlin, 1996. wKx S. K. Khanduja, A uniqueness problem in simple transcendental extensions of valued fields, Proc. Edinburgh Math. Soc. 37 Ž1993., 13᎐23. wMx S. MacLane, A construction for absolute values on polynomial rings, Trans. Amer. Math. Soc. 40 Ž1936., 363᎐395. wMax K. Mathiak, Bewertungen nichtkommutativer Korper, J. Algebra 48 Ž1977., 217᎐235. ¨ wMO1x M. Matignon and J. Ohm, A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104, No. 2 Ž1988., 392᎐402. wMO2x M. Matignon and J. Ohm, Simple transcendental extensions of valued fields, III. The uniqueness property, J. Math. Kyoto Uni¨ . 30 Ž1990., 347᎐366. wOx A. Ostrowski, Untersuchungen zur arithmetischen Theorie der Korper, Math. Z. 39 ¨ Ž1934., 269᎐404. wRx P. Ribenboim, ‘‘Theorie des Valuations,’’ Press Univ. Montreal, ´ ´ Montreal, ´ 1968.
Valuation Rings in Ore Extensions H. H. Brungs1 Uni¨ ersity of Alberta, Edmonton, Alberta T6G 2G1, Canada
and M. Schroder ¨ Uni¨ ersitat ¨ Duisburg, Lotharstrasse 65, D-47048, Duisburg, Germany Communicated by Susan Montgomery Received June 18, 1999
Extensions of valuation rings V of a skew field K are considered in the skew field F s K Ž x, . for a monomorphism of K. At least two such extensions exist if is an automorphism, but no extension may exist if is a monomorphism only. There exist extensions R Ž a. of V in F with xRŽ a. s aRŽ a. for every non-zero a in K if and only if V is invariant and is compatible with V. 䊚 2001 Academic Press
1. INTRODUCTION MacLane described in wMx the set of real-valued extensions of a Žrealvalued. valuation of a ground field F0 in a simple transcendental extension F1 s F0 Ž x .. There always exist infinitely many such extensions and some related results can be found in wRx and wBOx; see also wBx and wOx. Many other authors have considered the problem of how to extend a valuation on a field F0 to a simple transcendental extension F1 , for example, Matignon and Ohm wMO1, MO2x, Alexandru et al. wAPZx, and Khanduja wKx. Haase used the fact that a complete discrete valuation of the center F of a finite-dimensional division algebra D can be extended to D in order to compute the Brauer group of local fields. In contrast to the commutative situation, not every valuation of the center can be extended to a finite-dimensional division algebra; see for example wBGx. 1
The first author is supported by NSERC. 665 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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We say that the pair Ž K, V . is a valued skew field if K is a skew field with the subring V such that a g K _V implies ay1 g V; we also say that V is a valuation ring of K. Let be a monomorphism of K and F s K Ž x, . the corresponding Ore extension of K. We consider the set of extensions of V in F, i.e., the valuation rings R of F with R l K s V. The results are surprisingly different from the commutative situation. Even though at least two extensions of V exist in F if is an automorphism, see Theorem 2.2, an example is given, see 4.2, where is a monomorphism and V has no extension in F. In Section 3 the important condition that is compatible with V is discussed and V has extensions in F in this case. It is proved in 4.3 that an invariant rank-one valuation ring V has exactly two extensions in F s K Ž x, . for a non-compatible automorphism. In the special case V s K, an automorphism, all extensions of V are determined in 4.6. In Section 5 it is shown that for all a g K _ 04 there exist extensions R of V in F with xR s aR if and only if is compatible and V is invariant; see Theorem 5.1. Ultra products of extensions of a valuation ring V are considered in 5.3.
2. STANDARD EXTENSIONS Let K be a skew field with subring V such that a g K _V implies ay1 g V. We then say that the pair Ž K, V . is a valued skew field and that V is a valuation ring of K. We use UŽ V . to denote the group of units of V, and J Ž V . s V _UŽ V . is the maximal ideal of V. A valued skew field Ž F, R . is an extension of Ž K, V . if K : F is a skew field extension and R l K s V; we also say in this case that R is an extension of V in F. Let be a monomorphism of K and K w x, x s Ý x ia i ¬ a i g K 4 be the skew polynomial ring with ax s x Ž a. defining the multiplication where a g K. This ring is a right Ore domain; see wC, p. 54x, and hence F s K Ž x, . s K w x, x Sy1 , the skew field of quotients of K w x, x, exists, where S s K w x, x_ 04 . For V a valuation ring of K we investigate the set of extensions R of V in F, with F a simple Ore extension of K. The next well-known result shows that can be extended to an automorphism ˜ of a sub skew field K : K˜ of F such that F s K˜Ž x, ˜ .. 2.1. LEMMA. Gi¨ en a skew field K with monomorphism of K, then K˜ s D ⬁ns 0 x n Kxyn is a sub skew field of F s K Ž x, . and can be extended to an automorphism ˜ of K˜ such that F s K˜Ž x, ˜ .. Proof. If Tn denotes the skew field x n Kxyn , then Tn : Tnq1 , since x Kxyn s x n Ž x Ž K . xy1 . xyn s x nq1 Ž K . xyŽ nq1. : x nq1 KxyŽ nq1.. It follows that K˜ s jTn is a skew field contained in F. The mapping ˜ , with ˜ Ž x naxyn . s x n Ž a. xyn for a g K, is well defined and an automorphism n
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˜ Clearly, K˜ : F, and for Ýkis0 x ia˜i s 0, a˜i g K, ˜ there exists an n with of K. i . n n yn yn Ž a s x a x for a g K and all i. Hence, x Ý x a ˜i ˜i x s Ý x iai s 0 shows i i that a i s 0 for i s 0, . . . , k. It follows that F s K Ž x, . s K˜Ž x, ˜ .. If V is a valuation ring of K and if there exists an extension R of V in ˜ This suggests that conversely we F, then R l K˜ is an extension of V in K. might try to determine first the extensions V˜ of V in K˜ and then the extensions of the various V˜ in F s K˜Ž x, ˜ . where in this last step we know that ˜ is an automorphism. The next result shows that in the case where is an automorphism of K there always exist at least two extensions of V in F. In order to prove this theorem we consider skew power series rings and Laurent series rings. Let K be a skew field with automorphism . Then K ww x, xx s Ý⬁is0 x ia i ¬ a i g K 4 is the ring of skew power series which contains the right and left Ore set S s x i ¬ i G 04 . The multiplication in K ww x, xx is determined by ax s x Ž a. for all a in K. The localization K Ž Ž x, . . s Sy1 K w x, x s
½
⬁
Ý iGyn
x iai ai g K
5
is the skew field of skew Laurent series. For a non-zero element ␣ g K ŽŽ x, .., ␣ s Ý⬁iGyn x ia i , we define ¨ Ž ␣ . s k s min i ¬ a i / 04 and w Ž ␣ . s a k , the lowest non-zero coefficient of ␣ . 2.2. THEOREM.
Let be an automorphism of the skew field K and V a
¨ aluation ring of K. Then V has at least the two standard extensions R Ž1. and R Žy1. in F s K Ž x, .: The ¨ aluation ring R Ž1. of F is characterized as the
extension of V in F with the property that xa g J Ž R Ž1. . for all a g K, or equi¨ alently by the condition Ži.
xRŽ1. ; aRŽ1. for all 0 / a g K.
The ¨ aluation ring R Žy1. of F is characterized as the extension of V in F with the property that xy1 a g J Ž R Žy1. . for all a in K, or equi¨ alently by the condition Žii.
aRŽy1. ; xR Žy1. for all a g K.
Proof. Consider Fˆ s K ŽŽ x, .., the skew field of skew Laurent series which contains the skew field F s K Ž x, .. The subring Rˆ s V q ˆ It follows from the definition that xK ww x, xx is a valuation ring of F: ␣ g Fˆ is in Rˆ if ¨ Ž ␣ . ) 0 or if ¨ Ž ␣ . s 0 and w Ž ␣ . g V. If ¨ Ž ␣ . s 0 and Ž . w Ž ␣ . s a0 g K _V, then ay1 0 g V and from ␣ s a 0 1 y m , m g xK ww x, xx, it follows that ␣y1 s Ž1 q m q m2 q ⭈⭈⭈ . ay1 is an element in 0 ˆ Finally, if ¨ Ž ␣ . - 0, then ¨ Ž ␣y1 . ) 0 and ␣y1 g R. ˆ This shows that R.
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ˆ Rˆ. is a valued skew field. Let R Ž1. [ F l Rˆ and R Ž1. l K s Rˆ l K s V Ž F, follows; i.e., R Ž1. is an extension of V in F. Clearly, condition Ži. holds for R Ž1. since ay1 xR Ž1. ; R Ž1. for all 0 / a g K, and this condition, by the bijectivity of , is equivalent to the property that xa g J Ž R . for all 0 / a g K. To prove that property Ži. characterizes R Ž1. , let R be any extension of V in F with xa g J Ž R . and hence x na g J Ž R . for all n G 1 and a g K. It follows that all expressions of the form Ž1 q xc1 q ⭈⭈⭈ qx n c n . with c i g K are units in R. Every element ␣ in F can be written as ␣ s Ž a0 q xa1 q ⭈⭈⭈ qx na n . Ž b 0 q xb1 q ⭈⭈⭈ qx m bm . X
X
y1
X X. s x k a Ž 1 q xaX1 q ⭈⭈⭈ qx n aXnX . Ž 1 q xbX1 q ⭈⭈⭈ qx m bm
y1
for a i , aXi , bi , bXi , a g K and k g ⺪. We use the fact that is an automorphism to prove the second equation. It follows that ␣ is an element of R if and only if k ) 0 or k s 0 and a g V. This proves the uniqueness of R and hence R s R Ž1. . On the other hand, we can consider FˆŽy1. s K ŽŽ y, .. for y s xy1 and s y1 , which is a skew field that contains F. It also contains the valuation ring RˆŽy1. s V q yK ww y, xx, and R Žy1. [ RˆŽy1. l F is an extension of V in F with property Žii.. The proofs of the statements about R Žy1. are similar to the proofs given above for R Ž1. . 2.3. Remark. With the assumptions as in Theorem 2.2 let 0 / ␣ g F with ␣ s Ž x na0 q x nq1 a1 q ⭈⭈⭈ qx nqk a k .Ž x m b 0 q ⭈⭈⭈ qx mqt bt .y1 where a i , bj g K and a0 / 0 / b 0 . Then ␣ g R Ž1. if and only if either n ) m or . g V. n s m and ym Ž a0 by1 0 2.4. Remark. With the assumptions as in Theorem 2.2 let RX [ V q xK w x, x : F. Then RX is a subring of K w x, x and SX [ 1 q xa1 q x 2 a2 q ⭈⭈⭈ qx na n ¬ a1 , a2 , . . . , a n g K 4 is a right Ore system in RX with R Ž1. s RX SXy1. 2.5. COROLLARY.
Let be a monomorphism of the skew field K and V a X
˜ then V has at least two ¨ aluation ring of K. If V has an extension V˜ in K, distinct extensions
RXŽ1.
and
RXŽy1.
in F which are also extensions of V˜X .
3. COMPATIBILITY We consider in this section a condition on the monomorphism of the skew field K and the valuation ring V of K which guarantees that V has an extension in F s K Ž x, .. This condition was used in wMax and wBTx to
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construct valuation rings R that have no two-sided ideals / Ž0., J Ž R . and R, i.e., where R is nearly simple. 3.1. DEFINITION. The monomorphism of k is said to be compatible with the valued skew field Ž K, V . or we also say that is compatible with V if Ž V . : V and Ž J Ž V .. : J Ž V .. 3.2. LEMMA. The following conditions are equi¨ alent: Ži. Žii. Žiii. Živ.
is compatible with Ž K, V .; for all a in K, Ž a. is in V if and only if a is in V; for all a in K, Ž a. is in UŽ V . if and only if a is in UŽ V .; for all a in K, Ž a. is in J Ž V . if and only if a is in J Ž V ..
Proof. Ži. « Žii.. That a g V implies Ž a. g V follows directly from 3.1. If, conversely, Ž a. g V for a f V, then ay1 g J Ž V . and Ž ay1 . g J Ž V . follows by Ži.; this is a contradiction, since Ž a., Ž a.y1 g UŽ V .. Žii. « Žiii. If a g UŽ V ., with b its inverse, then Ž a. g UŽ V . with Ž b . as its inverse. Conversely, if Ž a. g UŽ V ., then there exists b g UŽ V . with Ž a. b s b Ž a. s 1. By Žii. we have a g V. If a g J Ž V ., then ay1 f V and b s Ž ay1 . f V, a contradiction. Žiii. « Živ. If a g J Ž V ., then 1 y a g UŽ V . and by Žiii. it follows that 1 y Ž a. g UŽ V . and hence that Ž a. g V. If Ž a. g UŽ V ., then a g UŽ V ., a contradiction. If, conversely, Ž a. g J Ž V ., then 1 y Ž a. s Ž1 y a. g UŽ V . and 1 y a g UŽ V . follows. If a g UŽ V ., then Ž a. g UŽ V ., a contradiction. Živ. ª Ži. If a g V, Ž a. f V, then Ž ay1 . g J Ž V .. Hence ay1 g Ž . J V , a contradiction. If a g J Ž V ., then Ž a. g J Ž V . by Živ.. 3.3. Remark. The compatibility of with Ž K, V . is also equivalent to the following condition: Žv. aV ª Ž a.V, for a g K, defines a strictly monotone mapping of the totally ordered lattice of cyclic right V-submodules of K into itself. 3.4. LEMMA. Let K˜ and ˜ be as in Lemma 2.1. If is compatible with Ž K, V ., then V˜ [ D nG 0 x n Vxyn is a ¨ aluation ring of K˜ that extends V to K˜ ˜ and ˜ is compatible with V. Proof. Since x n Vxyn s x nq1 Ž V . xyŽ nq1. : x nq1 VxyŽ nq1., and since for ˜ ␣ s x naxyn with a g K for some n, it follows that either ␣ or ␣y1 ␣ g K, ˜ Hence, V˜ is a valuation ring of K. ˜ If is contained in x n Vxyn : V. ␣ s x naxyn for a g V is in V˜ l K, then n Ž ␣ . s a g V and ␣ g V ˜ To follows, which shows V˜ l K s V; i.e., V˜ is an extension of V in K. ˜ show that ˜ is V-compatible we use the condition Žii. of 3.2 and the fact
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that an element ␣ g K˜ is in V˜ if and only if there exists an n G 0 with ˜ n Ž ␣ . g V. Hence, ␣ g V˜ implies ˜ n Ž ␣ . g V for some n and ˜ n Ž ˜ Ž ␣ .. ˜ then s ˜ Ž ˜ n Ž ␣ .. g V; ˜ Ž ␣ . g V˜ follows. Conversely, if ˜ Ž ␣ . g V, ˜ n Ž ˜ Ž ␣ .. s ˜ nq1 Ž ␣ . g V; ␣ g V˜ follows.
˜ see Lemma 2.1, is 3.5. Remarks. Ži. The automorphism ˜ of K, X ˜ ˜ compatible with a valuation ring V of K that extends V if and only if is ˜ compatible with V and V˜X s V. Žii. There exist examples for compatible with V and an extension V˜X of V in K˜ which is different from V˜ and is therefore not compatible with ˜ . Proof of Ži.. Assume that ˜ is compatible with the extension V˜X of V ˜ Let a g V : V˜X , and hence ˜ Ž a. s Ž a. g V˜X l K s V. Conin K. versely, if a g K and Ž a. g V, then ˜ Ž a. g V˜X , a g V˜X l K s V which shows that is compatible with V. Since ˜ is compatible with V˜ ŽLemma ˜ For ␣ g K˜ we have ␣ s x naxyn for 3.4., it remains to show that V˜X s V. some a g K, and some n G 0. This implies ˜ n Ž ␣ . s a and ␣ g V˜X if and only if a g V, using the compatibility of ˜ with V˜X . However, this condition is equivalent to the condition that requires ␣ to be an element ˜ hence V˜X s V˜ follows. of V, To prove Žii. we consider the following example. Let K [ Ž⺡Ž x 0 , x 1 , . . . . be the function field over the rationals in countably many indeterminates with the monomorphism defined by Ž x n . s x nq1 for n s 0, 1, 2, . . . . It follows that K˜ s ⺡Ž . . . , xy2 , xy1 , x 0 , x 1 , x 2 , . . . . with ˜ Ž x k . s x kq1 for k g ⺪. For K we consider the valuation ¨ which is defined on ⺡w x 0 , x 1 , x 2 , . . . x by ¨
Ž Ý rn , n , . . . Ž x 0n 0
1
0
x 1n1 . . . . . s min
½ Ýn
i
rn 0 , n1 , . . . / 0 ;
5
let V be the corresponding valuation ring. Then is compatible with V and V has infinitely many extensions in K˜; however, ˜ is compatible only with V˜ s DnG 0 x n Vxyn. It follows from the next result that an extension R of V exists in F that contains both x and xy1 if and only if is compatible with V. 3.6. THEOREM. Let be a monomorphism and V a ¨ aluation ring of the skew field K. There exists an extension R of V in F s K Ž x, . with x, xy1 g R if and only if is compatible with Ž K, V .. Proof. Assume that R is an extension of V in F with x g UŽ R .. For a g K we have ax s x Ž a. and hence a is a unit in V if and only if Ž a. s xy1 ax is a unit in R and therefore in V. It follows from Lemma 3.2Žiii. that is compatible with V.
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Conversely, if is compatible with V it follows from Lemma 3.4 that ˜ there exists an extension V˜ of V in K˜ such that ˜ is compatible with V. i ˜ ˜ ˜ 4 The set S s Ý x a i ¬ a i g V, Ýa i V s V is a right᎐left Ore system in the skew polynomial ring V˜w x, ˜ x such that R s V˜w x, ˜ x Sy1 is a valuation ring of F Žsee wBSchx.. It follows from the construction that x is in UŽ R .. To show that R is an extension of V we consider an element ␣ in the intersection R l K which certainly contains V. We have ␣ s ŽÝ x ia i .ŽÝ x i c i .y1 for Ý x i c i g S and Ý x ia i g V˜w x, ˜ x. It follows that Ý x ia i s ␣ Ý x i c i s Ý x i i Ž ␣ . c i and at least one of the coefficients c i , say c i 0 , is in UŽ V˜ . since Ý x i c i is in S. Then i 0 Ž ␣ . s a i 0 cy1 i 0 is an element in V˜ l K s V and ␣ g V follows by using Lemma 3.2Žii.. This shows that R l K s V. 4. COMPLETE SETS OF EXTENSIONS We discuss in this section cases where all extensions of Ž K, V . in F s K Ž x, . can be determined. After collecting conditions on and Ž K, V . that guarantee the existence of an extension R of V in F, we give an example of Ž K, V . and a monomorphism of K so that V has no extension in F. We also describe a case where the extensions R Ž1. and R Žy1. that occur in Theorem 2.2 comprise exactly the set of all extensions of V in F. The extensions of V in F for V s K and an automorphism are determined in Theorem 4.6. They are R Žy1. , F, and the f-adic valuation rings R f of F for an irreducible invariant polynomial f in K w x, x; here, R Ž1. s R x . In particular, V s K has exactly three extensions if is an automorphism of K so that no power n, n G 1, is an inner automorphism. 4.1. LEMMA. There exists an extension of V in K˜ and hence in F in any of the following cases: Ža. K is commutati¨ e; Žb. is an automorphism; Žc. is compatible with Ž K, V .. Proof. It follows from Theorem 2.2 that every extension of V in K˜ can be extended to an extension of V in F. So it remains to show that V can be extended to K˜ in all three cases Ža., Žb., and Žc.. If K is commutative, ˜ If is an automorphism, then then so is K˜ and V can be extended to K. ˜ Ž . K s K. If is compatible with K, V then V can be extended to K˜ by Lemma 3.4. The result in the case Žc. also follows from Theorem 3.6. We now construct an example of a valued skew field Ž K, V . and a monomorphism of K so that V has no extension in F s K Ž x, ..
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4.2. EXAMPLE. Let ⺡Ž t . be a simple transcendental extension of the rationals and the automorphism of ⺡Ž t . defined by Ž t . s ty1 . We take K [ ⺡Ž t .Ž y, . and show that a monomorphism of K is defined by Ž y . s y q yy1 and Ž t . s t Žand so Ž a. s a for all a g ⺡Ž t ... Let ␣ s Ý y ia i be an element in ⺡Ž t .w y, x. Then Ž ␣ . s ÝŽ y q yy1 . ia i / 0 for ␣ / 0 since the non-zero term y na n will not be cancelled in Ž ␣ . if y na n is the highest non-zero term in ␣ . It follows that induces a ⺡Ž t .-linear mapping from the ⺡Ž t . right vector space ⺡Ž t .w y, x into K and that this is a ring monomorphism since y1 s and aŽ y q yy1 . s ay q ayy1 s y Ž a. q yy1y1 Ž a. s Ž y q yy1 . Ž a. for a g ⺡Ž t .. However, ⺡Ž t .w y, x is a right Ore domain and the monomorphism can be extended to the skew field K of quotients of ⺡Ž t .w y, x. Let F [ K Ž x, .. In K we consider the valuation ring V which is uniquely determined as the extension of the t-adic valuation ring ⺡w t xŽ t . with the property that ya g J Ž V . for all a g ⺡Ž t .; here we use the fact that is an automorphism and apply Theorem 2.2. We now claim that V has no extension R in F. To prove this consider the element [ xyxy1 g F. It follows that
q y1 s y y1
y1
y1 y1
Ž ).
since q s xyx q xy x s x Ž y q y x s x Ž y . x s yxxy1 y1 y1 s y. Further, ty s yt and tx s xt imply t s xyy1 Ž xy1 tx . yxy1 s y1 y1 y1 y1 y1 xy tyx s xt x s t , i.e.,
y1 t s ty1 .
y1 . y1
y1
Ž )).
If we assume that R is an extension of V in F, then y g V ; R implies that q y1 g R by Ž).. By construction we have t g V ; R and ty1 f V, hence ty1 f R. It follows from Ž)). that f UŽ R ., hence g J Ž R . or y1 g J Ž R .. However, q y1 g R, g J Ž R . lead to the contradiction y1 g R, and similarly y1 g J Ž R . implies the contradiction g R. Hence, no extension of V exists in F. We prove in the next result that under certain circumstances the two extensions R Ž1. and R Žy1. described in Theorem 2.2 are the only extensions of V in F. We say that the valuation ring V of the skew field K is archimedean if for every a in K and every j g J Ž V . there exists an n G 0 with j na, aj n g J Ž V .. 4.3. THEOREM. Assume for the ¨ alued skew field Ž K, V . that is a non-compatible automorphism of K and that V is archimedean. Then R Ž1. and R Žy1. are exactly the extensions of V in F s K Ž x, ..
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Proof. Since is an automorphism, it follows from Theorem 2.2 that R Ž1. and R Žy1. are extensions of V in F. Conversely, we have to show that any extension R of V in F is equal to either R Ž1. or R Žy1. . Since is not compatible, condition Žiii. of Lemma 3.2 does not hold and either there exists an element u g UŽ V . with Ž u. f UŽ V . or there exists an element a g K _UŽ V . with Ž a. g UŽ V .. If in the first case Ž u. f J Ž V ., then uy1 g UŽ V . and Ž uy1 . g J Ž V ., i.e., we can assume that u g UŽ V . but Ž u. g J Ž V .. We can argue in a similar way in the second case and conclude that by the non-compatibility of we have either Ži. an element u g UŽ V . with Ž u. g J Ž V .; or Žii. an element j g J Ž V . with Ž j . g UŽ V .. Now we consider first the case x g R which by the non-compatibility of and Theorem 3.6 implies that x g J Ž R ., and then the case x f R which implies xy1 g J Ž R .. In the first case we will prove that xa g J Ž R . for any a g K, which in turn implies that R s R Ž1. by Theorem 2.2. Let a g K. In the case Ži. there exists an element u g UŽ V . with Ž u. g J Ž V . and hence for any n G 0 we obtain n
xa s uyn u n xa s uyn x Ž u . a. Since V is archimedean, it follows that Ž u. na g J Ž V . for n large enough, which implies that xa g J Ž R .. In the case Žii. there exists an element j g J Ž V . with Ž j . g UŽ V . and for any n G 0 we have xa s y1 Ž a . x s y1 Ž a . j n jyn x s y1 Ž a . j n x Ž j .
yn
,
.yn
where x g R, Ž j g UŽ V .. Since V is archimedean, it follows that y1 Ž a. j n g J Ž V . for n large enough. We conclude that xa g J Ž R . for any a g K and that R s R Ž1. by Theorem 2.2. We now assume that xy1 g R. Since is an automorphism we have F s K Ž x, . s K Ž xy1 , y1 .. We apply the case considered above and obtain xy1 a g J Ž R . for all a in K, since xy1 g R and y1 is not compatible with V. Then R s R Žy1. by Theorem 2.2. In the next result we characterize archimedean valuation rings in various ways. 4.4. PROPOSITION. The following conditions are equi¨ alent for a ¨ aluation ring V / K of a skew field K : Ža. V is archimedean; Žb. V is in¨ ariant and has rank one;
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Žc. there exists a ¨ aluation ¨ from K into Ž⺢, q. j ⬁4 with V s a g K ¬ ¨ Ž a. G 04 . We recall that V is called invariant if all its one-sided ideals are two-sided or, equivalently, if dVdy1 s V for all 0 / d g K. The valuation ring V has rank n if there exist exactly n completely prime ideals / Ž0. in V. A valuation from K to Ž⺢, q. j ⬁4 is a mapping from K into this set so that ¨ Ž a. s ⬁ if and only if a s 0; ¨ Ž ab. s ¨ Ž a. q ¨ Ž b . and ¨ Ž a q b . G min ¨ Ž a., ¨ Ž b .4 for a, b in K. In the proof of the proposition we show first that Žb. implies Žc.. If V is an invariant rank-one valuation ring of K, then aV ¬ 0 / a g K 4 is a group with aVbV s abV defining its operation. This group is isomorphic to a subgroup of Ž⺢, q., and a valuation as in Žc. can be defined. The condition Žc. implies Ža. since ¨ is a real-valued valuation. To show that V has rank one if V is archimedean, we assume on the contrary that there exists a completely prime ideal P with V > J Ž V . > P > Ž0. and we choose an element 0 / p g P and an element j g J Ž V ._ P. If for some n G 0 we have j n py1 s r g J Ž R ., then j n s rp g P, hence j g P, since P is completely prime; this is a contradiction. It remains to show that an archimedean valuation ring V is invariant and it is sufficient to prove that every principal right ideal of V is two-sided. Otherwise there exists an element 0 / a in V and an element r in V with ra f aV, hence raj s a for some element j g J Ž V .. We show next that we can assume r g UŽ V .. Otherwise we have r g J Ž V . and raj s a implies Ž r y 1. aj s raj y aj s aŽ1 y j . with r y 1, 1 y j g UŽ R .. Hence, Ž r y 1. ajŽ1 y j .y1 s a or uaw s a for u s r y 1 g UŽ R . and w s jŽ1 y j .y1 g J Ž R .. Then u naw n s a implies w nay1 s ay1 uyn f J Ž V . for all n G 0; this is a contradiction. We do not know the exact conditions on Ž K, V . and the automorphism such that the two standard extensions of V are exactly the extensions of V. We give an example of a non-invariant rank-2 valuation ring V with a non-compatible automorphism so that V has more than two extensions in F. 4.5. EXAMPLE. Consider K s ⺡Ž i .ŽŽ t, .., the skew field of skew Laurent series over ⺡Ž i ., i 2 s y1, with the automorphism equal to conjugations; i.e., Ž i . s yi. Further, let V s B q t⺡Ž i .ww t, xx where B s ⺪w i xŽ2yi. is the Ž2 y i .-adic valuation ring of ⺡Ž i ., and let X be the identity mapping of K. Then F s K Ž xX , X . contains an extension R of V with xX , xXy1 g R since X is compatible with V; see Theorem 3.6. However, F s K Ž x, . also for x s txX and with Ž a. s ty1 at for a g K. It follows that is not compatible with V, since for example Ž2 q i .y1 g V,
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but ŽŽ2 q i .y1 . s Ž2 y i .y1 f V. The three extensions R, R Žy1. , and R Ž1. are distinct, since xR s tR, xR Ž1. ; tR Ž1. and xR Žy1. > tR Žy1. , where R Ž1. and R Žy1. are the standard extensions described in Theorem 2.2 for V in F s K Ž x, .. We determine all extensions R of V with V s K in F s K Ž x, . for an automorphism. All one-sided ideals in T s K w x, w are principal and a two-sided ideal I / Ž0. of T has the form I s fT s Tf for an element f s x m f 1Ž x . a, with 0 / a in K, f 1Ž x . s x na n q ⭈⭈⭈ qxa1 q 1, and f 1T s Tf 1. We say that f is an invariant element in T. Hence, f 1Ž x . b s bf 1Ž x . for all b in K and xf 1Ž x . s f 1Ž x . x. It follows that i Ž b . s a i bay1 for a i / 0 and i Ž a i . s a i for all i. If no positive power of is an inner automorphism of K then x n T, n G 0, are the only non-zero ideals of T. Otherwise, there exists r G 1 minimal with r an inner automorphism, i.e., r Ž b . s ubuy1 for all b in K and a certain element u in K. It follows that f 1Ž x . s x r k a r k q ⭈⭈⭈ qx r a r q 1 with a r i in InvŽ K . s a g K ¬ Ž a. s a4 and uyi a r i in ZŽ K ., the center of K. Conversely, every element f s x m f 1Ž x . a, with f 1Ž x . of the above form, generates an ideal fT s Tf of T ; see wJ, Chap. Ix for related results. A maximal ideal fT s Tf is completely prime only if the element f is in addition irreducible in T ; i.e., fT is also a maximal right Žor left. ideal in T. If f is an invariant and irreducible element in T then S s T _ fT is a right Žand left. Ore system in T. To see this let g g T and s be in S and gT q sT s dT, g s dg 1 , s s ds1 , and g 1 u q s1¨ s 1 for elements d, g 1 , s1 , u, ¨ in T, d, s1 g S. If g 1 u f fT, then g 1 us1 s s1Ž1 y ¨ s1 . / 0 for us1 f fT. If g 1 u g fT, then s1¨ g 1 s g 1Ž1 y ug 1 . and ug 1 g fT, 1 y ug 1 g S. The ring R f [ TSy1 exists and is an extension of V s K in F. 4.6. THEOREM. The extensions of V s K in F s K Ž x, ., an automorphism, are F, R Žy1. , and the rings R f for any irreducible in¨ ariant element f in K w x, x. Proof. Let R be any extension of V s K in F. If xy1 is in J Ž R ., then xy1 a is in J Ž R . for all a in K and R s R Žy1. by Theorem 2.2. Otherwise we have x in R and hence T s K w x, x is a subring of R. Then J Ž R . l T s N is an ideal in T and N is completely prime since J Ž R . is completely prime. Hence, either N s Ž0., J Ž R . s Ž0., and R s F or N s fT s Tf for an irreducible, invariant element f in T. If s g S s T _ N, then sy1 g R and R = R f follows. If r s hgy1 g R_ R f , we can assume that h g S and g g N. Then h g UŽ R ., g g UŽ R ., and f g UŽ R .; this is a contradiction that shows R s R f . 4.7. EXAMPLES TO ILLUSTRATE THEOREM 4.6. Ži. Assume that no power n of an automorphism of K is an inner automorphism for n G 1.
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Then K w x, x has just one irreducible invariant polynomial, f s x, and V s K has exactly three extensions in F s K Ž x, .; F, R Ž1. , and R Žy1. . One possible example is K s ⺡Ž t r ¬ r g ⺡. with Ž t r . s t 2 r defining the automorphism. Žii. In the case where K q ⺓, the complex numbers, and is the usual conjugation, it follows that 2 s id is an inner automorphism and the invariant polynomials in ⺓w x, x s T have the form f s x m f 1Ž x . a for a in ⺓ and f 1Ž x . s Ý x 2 i c 2 i for c 2 i g ⺢, c0 s 1. The monic, invariant, irreducible polynomials in T are of the form f s x or f s x 2 q r with 0 - r g ⺢. The element g s x 2 y 1 s Ž x q i .Ž x q i . generates a prime ideal of T that is not completely prime. 4.8. Remark. If RX / F is an extension of V s K in F s K Ž x, . with an automorphism, then RX s RXrJ Ž RX . is a skew field finite dimensional over K. If V is a valuation ring of K and W is an extension of V in RX , then R s a g RX ¬ a q J Ž RX . g W 4 is an extension of V in F. This construction will always produce the extensions R Ž1. and R Žy1. of V and possibly additional extensions if K w x, x contains irreducible invariant monic elements f / x. However, all extensions of V are obtained in this way only in very special cases; see also the examples in Section 5.
5. EXTENSIONS R WITH DESIGNED VALUE FOR xR Let Ž K, V . be a valued skew field, a monomorphism of K, and F s K Ž x, .. Let W Ž K . s W Ž K, V . s Ž cV ¬ 0 / c g K 4 , F. be the value set of cyclic right V-modules / Ž0. in K, totally ordered by inclusion. If R is an extension of V in F, then W Ž K . can be embedded into W Ž F, R . by identifying cV with cR for 0 / c g K, and this embedding will be order preserving. Therefore it makes sense to discuss the position of an element ␣ in F U , i.e., of ␣ R, in relation to the original value set W Ž K .; we will only consider ␣ s x. There are two cases: first, xR s cR for some c g K _ 04 , and, second, xR / cR for all c g K. We consider the first case. If K and F s K Ž x . are commutative fields and V is a valuation ring of K, then there exists for every non-zero element a in K an extension R s R Ž a. of V in F with xR s aR. Now let Ž K, V . be a valued skew field and a monomorphism of K. Then there exists for a particular non-zero element a in K an extension R s R Ž a. of V in F s K Ž x, . with xR s aR if and only if ay1 xR s R. With xX [ ay1 x and the corresponding monomorphism X defined by X Ž c . s Ž acay1 . for c in K Žsuch that cxX s xX X Ž c .., we have F s K Ž xX , X . and xX R s R. By Theorem 3.6 such an extension R of V in F
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exists if and only if X is compatible with Ž K, V .. We can now prove the following result. 5.1. THEOREM. Let Ž K, V . be a ¨ alued skew field and a monomorphism of K. Then there exists for e¨ ery non-zero element a in K an extension R s R Ž a. of V in F s K Ž x, . with xR s aR if and only if is compatible with Ž K, V . and V is in¨ ariant. Proof. If is compatible with V and V is invariant, then for every 0 / a g K we have aVay1 s V, and for any c in K it follows that c is in V if and only if X Ž c . s Ž acay1 . is in V. Hence X is compatible with V, and an extension R of V exists in F with aR s xR by the observation before the theorem. To prove the converse, we observe first that is compatible with V by Theorem 3.6, since an extension R Ž1. of V exists with xR Ž1. s R Ž1.. In addition, X with X Ž c . s Ž acay1 . is compatible with V for all 0 / a, c in K by the above remarks. This means that c is in V if and only if Ž acay1 . is in V, if and only if acay1 is in V for all 0 / a in K ; the ring V is invariant. If V is an invariant valuation ring of K, then the monomorphism of K is compatible with V, if and only if X with X Ž c . s Ž acay1 . for c g K is compatible with V for every non-zero element a in K. Hence, the next result is a consequence of Theorem 5.1 and the remarks made before this theorem. 5.2. COROLLARY. tions are equi¨ alent:
Assume that V is in¨ ariant. Then the following condi-
Ži. is compatible with V. Žii. For e¨ ery non-zero element a in K there exists an extension R s R Ž a. of V in F s K Ž x, . with xR s aR. Žiii. There exists a non-zero element a0 in K and an extension R of V in F with xR s a0 R. We consider now the second case where xR / cR for all c in K. Then xR determines a cut C s ŽU, O . in the totally ordered set W Ž K . with U [ aV g W Ž K . ¬ aR : xR4 and O [ bV g W Ž K . ¬ xR ; bR4 .
Ž ).
Here, C ŽU, O . is a cut in W Ž K . in the general sense that ŽU, O . is a partition of W Ž K . so that U j O s W Ž K . and ␣ -  for all ␣ g U and all  g O. Further, ŽU, O . s ŽU X , OX . if and only if U s U X and O s OX .
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Given any cut C s ŽU, O . in W Ž K . and an extension R of V in F, we then say that x determines C Žat R . if and only if Ž). holds. We can ask the following question: Assuming that Ž K, V . is a valued skew field, is an automorphism of K, and C s ŽU, O . is a cut in W Ž K ., does there exist an extension R of V in F s K Ž x, . so that x determines C? If U s , the empty set, then R Ž1. is an extension of V so that x determines C s Ž , W Ž K .., and similarly x determines the cut ŽW Ž K ., . at R s R Žy1. Žsee Theorem 2.2.. We can therefore assume that both U / and O / for the cut C s ŽU, O . under consideration. If U has a maximal element aV, then x determines C at an extension R s R Ž a. of V with aR s xR. This extension problem was discussed in the first part of this section; we therefore assume also that U does not have a maximal element. Then there exists a cofinal subset ␥ i s c i V ¬ i g I 4 of U with I a totally ordered index set without maximal element and ␥ i - ␥ j if and only if i - j in I. Using this notation we have the following result. 5.3. PROPOSITION. Assume that there exists for e¨ ery ␥ i s c i V an extension R i of V in F with c i R i : xR i ; bR i for all i in I and all b g K _ 04 with bV g O. Then there exists an extension R s R C of V so that aR ; xR ; bR for all a, b in K with aV g U and bV g O; i.e., x determines the cut C at R. Proof. The extension R will be obtained by using a suitable ultra product of the R i . Let F be the filter on I consisting of all subsets of I that contain a subset of the form Si 0 s i g I ¬ i G i 0 4 for some i 0 in I. Let M be an ultra filter on I containing F. In the ring Ł i g I Ž F . i s Ž a i . i g I ¬ a i g F 4 we define an equivalence relation ‘‘; ’’ with Ž a i . ; Ž bi . if and only if the set i g I ¬ a i s bi 4 is an element in M. Then F˜ s Ł i g I Ž F . ir; is a skew field with elements wŽ a i .x corresponding to the equivalence class containing the element Ž a i .. The subset ˜ since R˜ s wŽ a i .x g F˜ ¬ a i g R i for all i4 is a valuation subring of F, wŽ a i .x g F˜_ R˜ implies that wŽ a i .x / wŽ0.x and i g I ¬ a i g R i 4 s S f M. Hence, S s i g I ¬ a i f R i 4 g M and S : i ¬ 0 / a i and ay1 g R i4 g M i ˜ and wŽ a i .xy1 g R. The mapping that assigns to a in F the element wŽ a.x in F˜ is an ˜ and we can assume that F˜ is an extension of F. If embedding of F into F, wŽ a.x s wŽ ri .x is an element in F˜ l K for a in K and ri in R i , then w I g I ¬ a s ri 4 g M and a g V follows; so R s R C [ R˜ l F is an extension of V in F. It remains to prove that c i 0 R : xR ; bR for all c i 0 , i 0 g I, and all b g K with bV g O. By assumption we have extensions R i of V with c i R i :
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xR i ; bR i for all i g I and b g K with bV g O. Hence, Ž1. Ž2.
xy1 c i g R i for all i in I, and xy1 b f R i for all i in I and b g K with bV g O.
Further, for each i 0 g I and i G i 0 we have c i 0V : c i V, and so elements d i g V : R i exist with c i 0 s c i d i . By Ž1. it follows that xy1 c i 0 s xy1 c i d i g ˜ where we can R i for i G i 0 , and therefore wŽ xy1 c i 0 .x s wŽ xy1 c i d i .x g R, y1 y1 wŽ .x define d i s 0 for i - i 0 . Hence, x c i 0 s x c i 0 g R˜ l F s R, and c i 0 R : xR follows for every i 0 g I. On the other hand, we have xy1 b g F with x -1 b f R i by Ž2. for all i g I. Hence, xy1 b s wŽ xy1 b .x f R˜ and therefore xy1 b f R˜ l F s R. It follows that Ž xy1 b .y1 g J Ž R . and xR ; bR for all b g K with bV g O. Combining Theorem 5.1 and Proposition 5.3 we obtain the following result: 5.4. COROLLARY. Let Ž K, V . be a ¨ alued field and an automorphism of K. Then there exists for e¨ ery cut C s ŽU, O . of W Ž V . an extension R s R C of V so that x determines C at R if V is in¨ ariant and is compatible with V. 5.5. EXAMPLE. Let K s ⺡Ž t r ¬ r g ⺡. and be the automorphism of K with Ž t r . s t 2 r as in Example 4.7Ži.. Let V be the t-adic valuation ring of K, and V is invariant, and compatible with V. Since W Ž V . is order isomorphic to the rational numbers, it follows by Corollary 5.4 that the cardinality of the set of extensions of V in F s K Ž x, . is greater than or equal to the cardinality of the set of real numbers. On the other hand, the extensions of V obtained as in the Remark 4.8 are just the extensions R Ž1. and R Žy1. . ACKNOWLEDGMENTS We thank the referee for several helpful comments.
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