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Free Subgroups and Free Subsemigroups of Division Rings
JOURNAL OF ALGEBRA ARTICLE NO.
184, 570]574 Ž1996.
0275
Free Subgroups and Free Subsemigroups of Division Rings Katsuo Chiba Niihama National College of Technology, Yagumo-Cho 7-1, Niihama 792, Japan Communicated by Susan Montgomery Received July 31, 1995
In w4x Lichtman raised a question whether the multiplicative group of a noncommutative division ring D contains a noncyclic free subgroup. The answer is positive when D is finite-dimensional over its center w2, Theorem 2.1x. Recently Reichstein and Vonessen showed that the multiplicative group of D contains a noncyclic free subgroup, if the center K of D is uncountable and there exists a non-central element a g D which is algebraic over K w8, Theorem 1x. In this note we prove the above result without the assumption of the existence of a non-central element a g D which is algebraic over K ŽTheorem 2.. On the other hand Klein raised a question whether the multiplicative semigroup of a noncommutative domain contains a noncommutative free subsemigroup and he proved that the multiplicative semigroup of the ring of polynomials in two central indeterminates over a noncommutative domain contains a noncommutative free subsemigroup w3x. Both of these questions are still open. Our main results are the following three theorems. THEOREM 1. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u, ¨ x the polynomial ring in two central indeterminates u, ¨ o¨ er D, and DŽ u, ¨ . the quotient di¨ ision ring of Dw u, ¨ x. Let a and b be elements of D such that ab y ba / 0 and write d12 s Ž ab y ba.y1 Ž a y u., d 21 s yb 2 Ž ab y ba.y1 Ž baby1 y u. g Dw u, ¨ x. Then the elements of S s Ž1 y f¨ d12 . Ž1 y g¨ d 21 .Ž1 q f¨ d12 .Ž1 q g¨ d 21 . N f, g g K w ¨ x _ 044 freely generates a free subgroup of the multiplicati¨ e group of DŽ u, ¨ .. THEOREM 2. Let D be a noncommutati¨ e di¨ ision ring with uncountable center K. Then the multiplicati¨ e group of D contains a noncyclic free subgroup whose cardinal number is the same as that of K. 570 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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THEOREM 3. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u x the polynomial ring in a central indeterminate u o¨ er D, and < K < the cardinal number of K. Then the multiplicati¨ e semigroup of Dw u x contains a noncommutati¨ e free subsemigroup on < K < free generators. Theorem 2 sharpens results of Makar]Limanov w6, Theoremx, and of Makar]Limanov and Malcolmson w7, Theoremx, and of w8, Theorem 1x. In w3x Klein asked whether the multiplicative semigroup of the polynomial ring in one central indeterminate over a noncommutative domain contains a noncommutative free subsemigroup. Theorem 3 gives an affirmative answer to the question for a special case. Let R be a ring. We denote by MnŽ R . the n = n matrix ring over R, by GLnŽ R . the group of units of MnŽ R ., and by e i j the matrix with 1 in the Ž i, j .-entry and 0 elsewhere. We denote by Rw ¨ x the polynomial ring in a central indeterminate ¨ over R and by Rww ¨ xx the formal power series ring over R. By the natural isomorphisms MnŽ Rw ¨ x. ( MnŽ R .w ¨ x and MnŽ Rww ¨ xx. ( MnŽ R .ww ¨ xx, we shall identify MnŽ Rw ¨ x. with MnŽ R .w ¨ x and MnŽ Rww ¨ xx. with MnŽ R .ww ¨ xx, respectively. Let V denote the ideal of Rww ¨ xx generated by ¨ and let X be a subset of Rww ¨ xx. We denote by X the closure of X in Rww ¨ xx with respect to V-adic topology on Rww ¨ xx. If R is a division ring, we denote by RŽ ¨ . and RŽŽ ¨ .. the quotient division rings of Rw ¨ x and Rww ¨ xx, respectively. There is a natural embedding RŽ ¨ . into RŽŽ ¨ .., so we shall think of RŽ ¨ . as a subring of RŽŽ ¨ ... Let G be a group, and Y a subset of G. If Y is a free set of generators of G, we also say that Y freely generates G or that G is free on Y. Let S be a set. We denote by < S < the cardinal number of S. For the proof of Theorem 1, we need the following. LEMMA 1. Let k be a commutati¨ e field and k w ¨ x the polynomial ring in a commutati¨ e indeterminate ¨ o¨ er k. Let A s 1 q f¨ e12 N f g k w ¨ x4 and B s 1 q g¨ e21 N g g k w ¨ x4 , the subgroups of GL2 Ž k w ¨ x.. Then the subgroup of GL2 Ž k w ¨ x. generated by S s ay1 by1 ab N a g A, a / 1, b g B, b / 14 is free on S. Proof. By w1, Lemma 5x the subgroup of GL2 Ž k w ¨ x. generated by A j B is the free product A) B of the groups A and B. By w5, Sect. 4.1, Problem 24, p. 196x the subgroup of GL2 Ž k w ¨ x. generated by S is free on S. This completes the proof. Proof of Theorem 1. We may regard Dw ¨ x as a right K w u, ¨ x-module by putting x ? f Ž u, ¨ . s xf Ž a, ¨ . for x g Dw ¨ x and f Ž u, ¨ . g K w u, ¨ x. Since Dw u, ¨ x s D mK K w u, ¨ x, Dw ¨ x becomes a left Dw u, ¨ x-module in the natural way. Let R s K ² d12 , d 21 , ¨ : be the K-subalgebra of Dw u, ¨ x generated by d12 , d 21 , and ¨ and M s 1 ? K w ¨ x q b ? K w ¨ x a right K w ¨ x-submodule of Dw ¨ x generated by K w ¨ x-linearly independent elements 1 and b. Since
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rM ; M for all r g R, the map w : R ª M2 Ž K w ¨ x. defined by Ž r1, rb . s Ž1, b . w Ž r . is a ring homomorphism such that w Ž d12 . s e12 , w Ž d 21 . s e21 , and w Ž ¨ . s ¨ . Let V1 be the ideal of Dw u, ¨ x generated by ¨ , V2 the ideal of R generated by ¨ , V3 the ideal of M2 Ž K w ¨ x. generated by ¨ , and V the ideal of Dw u xww ¨ xx generated by ¨ . Then V2n s V1n l R for all positive integers n and w Ž V2 . s V3 ; hence the V2-adic completion of R is naturally isomorphic to R, the closure of R in Dw u xww ¨ xx with respect to V-adic topology on Dw u xww ¨ xx, and the V3-adic completion of M2 Ž K w ¨ x. s M2 Ž K .w ¨ x is M2 Ž K .ww ¨ xx s M2 Ž K ww ¨ xx., therefore w extends to a ring homomorphism w : R ª M2 Ž K ww ¨ xx. s M2 Ž K .ww ¨ xx. By Lemma 1, w Ž S . s Ž1 y f¨ e12 .Ž1 y g¨ e21 .Ž1 q f¨ e12 .Ž1 q g¨ e21 . N f, g g K w ¨ x _ 044 freely generates a free subgroup of GL2 Ž K w ¨ x.. Since the elements of S are invertible in R, S freely generates a free subgroup of the unit group of R. Since DŽ u, ¨ . ; DŽ u.ŽŽ ¨ .. and R ; Dw u xww ¨ xx ; DŽ u.ŽŽ ¨ .., the inverses of the elements of S in R are contained in DŽ u, ¨ .. Thus the subgroup of the multiplicative group of DŽ u, ¨ . generated by S is free on S. This completes the proof. Proof of Theorem 2. By Theorem 1, it suffices to show that D contains a noncommutative division ring E and two central elements u, ¨ such that Ž1. u and ¨ are algebraically independent over E, and Ž2. the cardinality of the center of E is equal to the cardinality of the center of D. By hypothesis D / K, there are two elements a, b in D such that ab y ba / 0. Let D 1 be the subdivision ring of D generated by a and b, K 1 the center of D 1 , K 1Ž K . the subfield of D generated by K 1 and K, and u, ¨ 4 j ¨ i N i g I 4 a transcendental basis of K 1Ž K . over K 1 , where I is the index set. Since K 1 is finite or countable and K is uncountable, we have < I < s < K <. We show that the power products W s u p ¨ q ¨ i1r1 ¨ i2r 2 ??? ¨ irnn N p G 0, q G 0, ri G 0, i k g I 4 in u, ¨ , and the ¨ i ’s are linearly independent over D 1. If they are linearly dependent over D 1 , take a minimal linearly dependent subset w 1 , w 2 , . . . , wt g W say, where w 1 is linearly dependent on w 2 , w 3 , . . . , wt : w 1 s Ýt2 d i wi , where d i g D 1. We may assume d 2 f K 1. Then we obtain d g D 1 such that dd2 y d 2 d / 0. Since dŽÝt2 d i wi . y ŽÝt2 d i wi . d s Ýt2 Ž dd i y d i d . wi s 0, we obtain a linearly dependent set w 2 , w 3 , . . . , wt . This contradicts the minimality of w 1 , w 2 , . . . , wt . Let E be the division ring generated by D 1 and all ¨ i , i g I. Then the division ring E and two central elements u, ¨ satisfy the conditions Ž1., Ž2.. This completes the proof. For the proof of Theorem 3, we need the following lemma, which is a generalization of Reichstein and Vonessen’s results w8, Proposition 8x. LEMMA 2. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u x the polynomial ring in a central indeterminate u o¨ er D, and DŽ u. the quotient
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di¨ ision ring of Dw u x and < K < the cardinal number of K. If there is a non-central element a of D which is algebraic o¨ er K, then the multiplicati¨ e group of DŽ u. contains a noncyclic free subgroup on < K < free generators which are contained in Dw u x. Proof. If K is uncountable, the results follow from Theorem 2, since the multiplicative group of D contains a free group on a subset X of D such that < X < s < K <. It is clear that the multiplicative semigroup of Dw u x generated by X is free. So we assume that K is finite or countable. By Reichstein and Vonessen’s results w8, Proposition 8x the multiplicative group of DŽ u. contains a free group on x, y4 , where x, y g Dw u x. By w5, Sect. 1.4, Problem 12, p. 43x the subset x n y n N n s 1, 2, 3, . . . 4 of Dw u x freely generates a free subgroup of the multiplicative group of DŽ u.. This completes the proof. Proof of Theorem 3. If K is uncountable, then the multiplicative semigroup of Dw u x contains a free semigroups on < K < free generators, as we showed in the proof of Lemma 2. Thus it suffices to show that Dw u x contains a free subsemigroup on countably free generators if K is finite or countable. By Lemma 2 we may assume that the non-central elements of D are transcendental over its center K. Let a and b be elements of D such that ab y ba / 0 and put d12 s Ž ab y ba.y1 Ž a y u., d 21 s yb 2 Ž ab y ba.y1 Ž baby1 y u.. We may regard D as a right K w u x-module by putting x ? f Ž u. s xf Ž a. for x g D and f Ž u. g K w u x. Since Dw u x s D mK K w u x, D becomes a left Dw u x-module in the natural way. Let R be the K-subalgebra of Dw u x generated by d12 , d 21 , and u and let N s 1 ? K w u x q b ? K w u x be a right K w u x-submodule of D generated by 1 and b. Since a is transcendental over K and ab y ba / 0, the elements 1 and b are linearly independent over K w u x. Since rN ; N for all r g R, the map c : R ª M2 Ž K w u x. defined by Ž r1, rb . s Ž1, b . c Ž r . is a ring homomorphism such that c Ž d12 . s e12 , c Ž d 21 . s e21 , and c Ž u. s u. Let S s Ž1 y fud12 .Ž1 y gud 21 .Ž1 q fud12 .Ž1 q gud 21 . N f, g g K w u x _ 044 . By Lemma 1, c Ž S . s Ž1 y fue12 .Ž1 y gue21 .Ž1 q fue12 .Ž1 q gue21 . N f, g g K w u x _ 044 freely generates a free subgroup of GL2 Ž K w u x.. Hence S freely generates a free subsemigroup in the multiplicative semigroup of Dw u x and it is clear that S is countable. This completes the proof.
REFERENCES 1. G. M. Bergman, Ordering coproducts of groups and semigroups, J. Algebra 133 Ž1990., 313]339. 2. J. Z. Goncalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Canad. Math. Bull. 27 Ž1984., 365]370.
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KATSUO CHIBA
3. A. A. Klein, Free semigroups of domains, Proc. Amer. Math. Soc. 116 Ž1992., 339]341. 4. A. I. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 Ž1977., 15]16. 5. W. Magnus, A. Karrass, and D. Solitar, ‘‘Combinatorial Group Theory,’’ 2nd rev. ed., Dover, New York, 1976. 6. L. Makar]Limanov, On free subsemigroups of skew fields, Proc. Amer. Math. Soc. 91 Ž1984., 189]191. 7. L. Makar]Limanov and P. Malcolmson, Words periodic over the center of a division ring, Proc. Amer. Math. Soc. 93 Ž1985., 590]592. 8. Z. Reichstein and N. Vonessen, Free subgroups of division algebras, Comm. Algebra 23 Ž1995., 2181]2185.
184, 570]574 Ž1996.
0275
Free Subgroups and Free Subsemigroups of Division Rings Katsuo Chiba Niihama National College of Technology, Yagumo-Cho 7-1, Niihama 792, Japan Communicated by Susan Montgomery Received July 31, 1995
In w4x Lichtman raised a question whether the multiplicative group of a noncommutative division ring D contains a noncyclic free subgroup. The answer is positive when D is finite-dimensional over its center w2, Theorem 2.1x. Recently Reichstein and Vonessen showed that the multiplicative group of D contains a noncyclic free subgroup, if the center K of D is uncountable and there exists a non-central element a g D which is algebraic over K w8, Theorem 1x. In this note we prove the above result without the assumption of the existence of a non-central element a g D which is algebraic over K ŽTheorem 2.. On the other hand Klein raised a question whether the multiplicative semigroup of a noncommutative domain contains a noncommutative free subsemigroup and he proved that the multiplicative semigroup of the ring of polynomials in two central indeterminates over a noncommutative domain contains a noncommutative free subsemigroup w3x. Both of these questions are still open. Our main results are the following three theorems. THEOREM 1. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u, ¨ x the polynomial ring in two central indeterminates u, ¨ o¨ er D, and DŽ u, ¨ . the quotient di¨ ision ring of Dw u, ¨ x. Let a and b be elements of D such that ab y ba / 0 and write d12 s Ž ab y ba.y1 Ž a y u., d 21 s yb 2 Ž ab y ba.y1 Ž baby1 y u. g Dw u, ¨ x. Then the elements of S s Ž1 y f¨ d12 . Ž1 y g¨ d 21 .Ž1 q f¨ d12 .Ž1 q g¨ d 21 . N f, g g K w ¨ x _ 044 freely generates a free subgroup of the multiplicati¨ e group of DŽ u, ¨ .. THEOREM 2. Let D be a noncommutati¨ e di¨ ision ring with uncountable center K. Then the multiplicati¨ e group of D contains a noncyclic free subgroup whose cardinal number is the same as that of K. 570 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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THEOREM 3. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u x the polynomial ring in a central indeterminate u o¨ er D, and < K < the cardinal number of K. Then the multiplicati¨ e semigroup of Dw u x contains a noncommutati¨ e free subsemigroup on < K < free generators. Theorem 2 sharpens results of Makar]Limanov w6, Theoremx, and of Makar]Limanov and Malcolmson w7, Theoremx, and of w8, Theorem 1x. In w3x Klein asked whether the multiplicative semigroup of the polynomial ring in one central indeterminate over a noncommutative domain contains a noncommutative free subsemigroup. Theorem 3 gives an affirmative answer to the question for a special case. Let R be a ring. We denote by MnŽ R . the n = n matrix ring over R, by GLnŽ R . the group of units of MnŽ R ., and by e i j the matrix with 1 in the Ž i, j .-entry and 0 elsewhere. We denote by Rw ¨ x the polynomial ring in a central indeterminate ¨ over R and by Rww ¨ xx the formal power series ring over R. By the natural isomorphisms MnŽ Rw ¨ x. ( MnŽ R .w ¨ x and MnŽ Rww ¨ xx. ( MnŽ R .ww ¨ xx, we shall identify MnŽ Rw ¨ x. with MnŽ R .w ¨ x and MnŽ Rww ¨ xx. with MnŽ R .ww ¨ xx, respectively. Let V denote the ideal of Rww ¨ xx generated by ¨ and let X be a subset of Rww ¨ xx. We denote by X the closure of X in Rww ¨ xx with respect to V-adic topology on Rww ¨ xx. If R is a division ring, we denote by RŽ ¨ . and RŽŽ ¨ .. the quotient division rings of Rw ¨ x and Rww ¨ xx, respectively. There is a natural embedding RŽ ¨ . into RŽŽ ¨ .., so we shall think of RŽ ¨ . as a subring of RŽŽ ¨ ... Let G be a group, and Y a subset of G. If Y is a free set of generators of G, we also say that Y freely generates G or that G is free on Y. Let S be a set. We denote by < S < the cardinal number of S. For the proof of Theorem 1, we need the following. LEMMA 1. Let k be a commutati¨ e field and k w ¨ x the polynomial ring in a commutati¨ e indeterminate ¨ o¨ er k. Let A s 1 q f¨ e12 N f g k w ¨ x4 and B s 1 q g¨ e21 N g g k w ¨ x4 , the subgroups of GL2 Ž k w ¨ x.. Then the subgroup of GL2 Ž k w ¨ x. generated by S s ay1 by1 ab N a g A, a / 1, b g B, b / 14 is free on S. Proof. By w1, Lemma 5x the subgroup of GL2 Ž k w ¨ x. generated by A j B is the free product A) B of the groups A and B. By w5, Sect. 4.1, Problem 24, p. 196x the subgroup of GL2 Ž k w ¨ x. generated by S is free on S. This completes the proof. Proof of Theorem 1. We may regard Dw ¨ x as a right K w u, ¨ x-module by putting x ? f Ž u, ¨ . s xf Ž a, ¨ . for x g Dw ¨ x and f Ž u, ¨ . g K w u, ¨ x. Since Dw u, ¨ x s D mK K w u, ¨ x, Dw ¨ x becomes a left Dw u, ¨ x-module in the natural way. Let R s K ² d12 , d 21 , ¨ : be the K-subalgebra of Dw u, ¨ x generated by d12 , d 21 , and ¨ and M s 1 ? K w ¨ x q b ? K w ¨ x a right K w ¨ x-submodule of Dw ¨ x generated by K w ¨ x-linearly independent elements 1 and b. Since
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rM ; M for all r g R, the map w : R ª M2 Ž K w ¨ x. defined by Ž r1, rb . s Ž1, b . w Ž r . is a ring homomorphism such that w Ž d12 . s e12 , w Ž d 21 . s e21 , and w Ž ¨ . s ¨ . Let V1 be the ideal of Dw u, ¨ x generated by ¨ , V2 the ideal of R generated by ¨ , V3 the ideal of M2 Ž K w ¨ x. generated by ¨ , and V the ideal of Dw u xww ¨ xx generated by ¨ . Then V2n s V1n l R for all positive integers n and w Ž V2 . s V3 ; hence the V2-adic completion of R is naturally isomorphic to R, the closure of R in Dw u xww ¨ xx with respect to V-adic topology on Dw u xww ¨ xx, and the V3-adic completion of M2 Ž K w ¨ x. s M2 Ž K .w ¨ x is M2 Ž K .ww ¨ xx s M2 Ž K ww ¨ xx., therefore w extends to a ring homomorphism w : R ª M2 Ž K ww ¨ xx. s M2 Ž K .ww ¨ xx. By Lemma 1, w Ž S . s Ž1 y f¨ e12 .Ž1 y g¨ e21 .Ž1 q f¨ e12 .Ž1 q g¨ e21 . N f, g g K w ¨ x _ 044 freely generates a free subgroup of GL2 Ž K w ¨ x.. Since the elements of S are invertible in R, S freely generates a free subgroup of the unit group of R. Since DŽ u, ¨ . ; DŽ u.ŽŽ ¨ .. and R ; Dw u xww ¨ xx ; DŽ u.ŽŽ ¨ .., the inverses of the elements of S in R are contained in DŽ u, ¨ .. Thus the subgroup of the multiplicative group of DŽ u, ¨ . generated by S is free on S. This completes the proof. Proof of Theorem 2. By Theorem 1, it suffices to show that D contains a noncommutative division ring E and two central elements u, ¨ such that Ž1. u and ¨ are algebraically independent over E, and Ž2. the cardinality of the center of E is equal to the cardinality of the center of D. By hypothesis D / K, there are two elements a, b in D such that ab y ba / 0. Let D 1 be the subdivision ring of D generated by a and b, K 1 the center of D 1 , K 1Ž K . the subfield of D generated by K 1 and K, and u, ¨ 4 j ¨ i N i g I 4 a transcendental basis of K 1Ž K . over K 1 , where I is the index set. Since K 1 is finite or countable and K is uncountable, we have < I < s < K <. We show that the power products W s u p ¨ q ¨ i1r1 ¨ i2r 2 ??? ¨ irnn N p G 0, q G 0, ri G 0, i k g I 4 in u, ¨ , and the ¨ i ’s are linearly independent over D 1. If they are linearly dependent over D 1 , take a minimal linearly dependent subset w 1 , w 2 , . . . , wt g W say, where w 1 is linearly dependent on w 2 , w 3 , . . . , wt : w 1 s Ýt2 d i wi , where d i g D 1. We may assume d 2 f K 1. Then we obtain d g D 1 such that dd2 y d 2 d / 0. Since dŽÝt2 d i wi . y ŽÝt2 d i wi . d s Ýt2 Ž dd i y d i d . wi s 0, we obtain a linearly dependent set w 2 , w 3 , . . . , wt . This contradicts the minimality of w 1 , w 2 , . . . , wt . Let E be the division ring generated by D 1 and all ¨ i , i g I. Then the division ring E and two central elements u, ¨ satisfy the conditions Ž1., Ž2.. This completes the proof. For the proof of Theorem 3, we need the following lemma, which is a generalization of Reichstein and Vonessen’s results w8, Proposition 8x. LEMMA 2. Let D be a noncommutati¨ e di¨ ision ring with center K, Dw u x the polynomial ring in a central indeterminate u o¨ er D, and DŽ u. the quotient
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di¨ ision ring of Dw u x and < K < the cardinal number of K. If there is a non-central element a of D which is algebraic o¨ er K, then the multiplicati¨ e group of DŽ u. contains a noncyclic free subgroup on < K < free generators which are contained in Dw u x. Proof. If K is uncountable, the results follow from Theorem 2, since the multiplicative group of D contains a free group on a subset X of D such that < X < s < K <. It is clear that the multiplicative semigroup of Dw u x generated by X is free. So we assume that K is finite or countable. By Reichstein and Vonessen’s results w8, Proposition 8x the multiplicative group of DŽ u. contains a free group on x, y4 , where x, y g Dw u x. By w5, Sect. 1.4, Problem 12, p. 43x the subset x n y n N n s 1, 2, 3, . . . 4 of Dw u x freely generates a free subgroup of the multiplicative group of DŽ u.. This completes the proof. Proof of Theorem 3. If K is uncountable, then the multiplicative semigroup of Dw u x contains a free semigroups on < K < free generators, as we showed in the proof of Lemma 2. Thus it suffices to show that Dw u x contains a free subsemigroup on countably free generators if K is finite or countable. By Lemma 2 we may assume that the non-central elements of D are transcendental over its center K. Let a and b be elements of D such that ab y ba / 0 and put d12 s Ž ab y ba.y1 Ž a y u., d 21 s yb 2 Ž ab y ba.y1 Ž baby1 y u.. We may regard D as a right K w u x-module by putting x ? f Ž u. s xf Ž a. for x g D and f Ž u. g K w u x. Since Dw u x s D mK K w u x, D becomes a left Dw u x-module in the natural way. Let R be the K-subalgebra of Dw u x generated by d12 , d 21 , and u and let N s 1 ? K w u x q b ? K w u x be a right K w u x-submodule of D generated by 1 and b. Since a is transcendental over K and ab y ba / 0, the elements 1 and b are linearly independent over K w u x. Since rN ; N for all r g R, the map c : R ª M2 Ž K w u x. defined by Ž r1, rb . s Ž1, b . c Ž r . is a ring homomorphism such that c Ž d12 . s e12 , c Ž d 21 . s e21 , and c Ž u. s u. Let S s Ž1 y fud12 .Ž1 y gud 21 .Ž1 q fud12 .Ž1 q gud 21 . N f, g g K w u x _ 044 . By Lemma 1, c Ž S . s Ž1 y fue12 .Ž1 y gue21 .Ž1 q fue12 .Ž1 q gue21 . N f, g g K w u x _ 044 freely generates a free subgroup of GL2 Ž K w u x.. Hence S freely generates a free subsemigroup in the multiplicative semigroup of Dw u x and it is clear that S is countable. This completes the proof.
REFERENCES 1. G. M. Bergman, Ordering coproducts of groups and semigroups, J. Algebra 133 Ž1990., 313]339. 2. J. Z. Goncalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Canad. Math. Bull. 27 Ž1984., 365]370.
574
KATSUO CHIBA
3. A. A. Klein, Free semigroups of domains, Proc. Amer. Math. Soc. 116 Ž1992., 339]341. 4. A. I. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 Ž1977., 15]16. 5. W. Magnus, A. Karrass, and D. Solitar, ‘‘Combinatorial Group Theory,’’ 2nd rev. ed., Dover, New York, 1976. 6. L. Makar]Limanov, On free subsemigroups of skew fields, Proc. Amer. Math. Soc. 91 Ž1984., 189]191. 7. L. Makar]Limanov and P. Malcolmson, Words periodic over the center of a division ring, Proc. Amer. Math. Soc. 93 Ž1985., 590]592. 8. Z. Reichstein and N. Vonessen, Free subgroups of division algebras, Comm. Algebra 23 Ž1995., 2181]2185.