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Characterizing when the Rings R and R[T] Are Integrally Closed via Linear Equations over R
Journal of Algebra 217, 628᎐649 Ž1999. Article ID jabr.1998.7821, available online at http:rrwww.idealibrary.com on
Characterizing when the Rings R and Rw T x Are Integrally Closed via Linear Equations over R A. DeFrancisco-Iribarren* and J. A. Hermida-Alonso† Departamento de Matematicas, Uni¨ ersidad de Leon, ´ ´ 24071 Leon, ´ Spain Communicated by D. A. Buchsbaum Received July 17, 1998
INTRODUCTION Let R be a commutative ring with an identity element. This paper is devoted to study when R and the polynomial ring Rw T x are integrally closed in terms of the systems of linear equations over R. Denote by T Ž R . the total quotient ring of R Ži.e., the ring of fractions of R with respect to the multiplicative set of all nonzero divisors of R .. The ring R is integrally closed if every x g T Ž R . which is integral over R belongs to R. If R is an integrally closed domain then so is the polynomial ring an one indeterminate Rw T x. On the other hand if R is an integrally closed ring with nonzero nilradical then Rw T x is never integrally closed Žthe element arT is integral over, but does not belong to, Rw T x for every nonzero nilpotent a in R .. The question is: When is the polynomial ring Rw T x integrally closed? Necessary conditions for Rw T x to be integrally closed are that R is reduced and integrally closed. However these conditions are not sufficient, see Žw6, p. 187x.. T. Akiba w1x proves that if R is a reduced ring such that for each maximal ideal ᒊ of R the local ring R ᒊ is an integrally closed domain then Rw T x is integrally closed. In the same paper it is proven that if R is a reduced integrally closed with Min R compact then Rw T x is integrally closed if and only if T Ž R . is a Von Neumann regular ring. T. Lucas gave necessary and sufficient conditions in order that the polynomial ring Rw T x be integrally closed. In w7x it is shown that Rw T x is integrally closed if and only if R is integrally closed in T Ž Rw T x.. In w8x it is *, † Partially supported by DGICYT PB95-0603-C02 and Junta Castilla y Leon ´ LE09r95. 628 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
LINEAR EQUATIONS
629
shown that Rw T x is integrally closed if and only if R is integrally closed in the ring of finite fractions of R. Let Ž S . : A. x s b a system of linear equations over R. If Ž S . has a solution in R then Up Ž A . s Up Ž A < b . for p G 0, where Up Ž A. denotes the pth-determinantal ideal of A and Ž A < b . denotes the augmented matrix of Ž S .. In general, the converse is not true. In w3x it is proven that the converse is true for any system of linear equations over R if and only if R is a Prufer ¨ ring. Next we expose the main results of this paper. The system Ž P . : a. x s b given by a1 . x s b1 a2 . x s b 2 Ž P. : .. .
¡
~
¢a
m.x
s bm
is called a proportionality. In Section 2 we prove that R is an integrally closed ring if and only if any proportionality Ž P . : a. x s b such that U1Ž a. contains a nonzero divisor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. We also characterize the rings R such that R ᒊ is an integrally closed domain for each maximal ideal ᒊ of R in terms of the all proportionalities over R. Section 3 is devoted to solve the question, when is Rw T x an integrally closed ring? In fact we prove that Rw T x is inegrally closed if and only if R is reduced and any proportionality Ž P . : a. x s b such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. As a consequence of the above characterizations we prove that if R is a reduced ring satisfying Property A then Rw T x is integrally closed if and only if R is integrally closed. Akiba w1x established this result using a different technique. Finally we prove that if Min R is quasi-compact then Rw T x is integrally closed if and only if R ᒊ is a domain integrally closed for every maximal ideal ᒊ of R. 1. OVERDETERMINED SYSTEMS OF LINEAR EQUATIONS Let Ž S . : A. x s b be a system of linear equations where A s Ž a i j . is a Ž m = n.-matrix with entries in R, b s Ž b1 , . . . , bm . t is a column vector with entries in R and x s Ž x 1 , . . . , x n . t is the column of indeterminates.
630
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
If f : R ª R⬘ is a ring homomorphism we denote by Ž f Ž S .. the R⬘system given by Ž f Ž S .. : f Ž A.. x s f Ž b . where f Ž A. s Ž f Ž a i j .. and f Ž b . s Ž f Ž b1 ., . . . , f Ž bm .. t. If R is contained on R⬘ and f is the inclusion homomorphism we say that Ž S . has a solution in R⬘ when the R⬘-system Ž f Ž S .. has a solution Žin R⬘.. We denote by rank R Ž A. the rank of A; i.e., the largest nonnegative integer p such that Up Ž A. / 0 where Up Ž A. is the ideal of R generated by all p = p-minors of A. If Ž S . : A. x s b has a solution in R then we have Up Ž A . s Up Ž A N b .
for p G 0,
where Ž A N b . is the augmented matrix of the system Ži.e., the Ž m = Ž n q 1..-matrix obtained from A by adding the column matrix b .. We say that Ž S . : A. x s b, where A is an Ž m = n.-matrix, is an overdetermined system if rank R Ž A. s n. The particular case
¡a . x s b 1
1
~
a2 . x s b 2 Ž P. : .. .
¢a
m.x
s bm
is called a proportionality and it is denoted by Ž P . : a. x s b. The next result is known as McCoy’s theorem, see w10, 11, p. 63x. THEOREM 1.1. The o¨ erdetermined homogeneous system Ž Sh . : A. x s 0 where A is a Ž m = n.-matrix uniquely has the tri¨ ial solution if and only if UnŽ A. is a faithful ideal Ž i.e., Ž0 : UnŽ A.. s 0.. Remark. Let J be an ideal of R. If R is a Noetherian ring then J is a faithful ideal if and only if grR J G 1, where grR J is the classical grade of J on R; i.e., the upper bound of the lengths of all R-sequences of elements of J. W. V. Vasconcelos in w14x constructs a faithful finitely generated ideal J in a commutative ring R such that every element of J is a zero divisor in R. The true grade or polynomial grade of J on R, notion due to M. Hochster, is defined by Gr R J s lim grRwT1 , . . . , T n x J . R w T1 , . . . , Tn x . nª⬁
Note that Gr R J G 1 if and only if J is a faithful ideal and when R is a Noetherian ring Gr R J s grR J. D. W. Sharpe w13x and J. A. Hermida-Alonso w4, 5x give sufficient conditions involving the true grade of the ideals Up Ž A. to assure that Ž S . has a solution in R.
631
LINEAR EQUATIONS
LEMMA 1.2. Let Ž S . : A. x s b an o¨ erdetermined system such that rank R Ž A. s rank R Ž A N b . s r. Then Ži. If Ur Ž A. contains a nonzero di¨ isor then Ž S . has a solution Ž necessarily unique. in the total quotient ring T Ž R . of R. Žii. If Ur Ž A. is a faithful ideal then Ž S . has a solution Ž necessarily unique. in the total quotient ring T Ž Rw T x. of Rw T x. Proof. Ži. Let be the canonical ring homomorphism from R to T Ž R .. Consider the T Ž R .-system
Ž Ž S . . : Ž A. . x s Ž b . . Since U Ž A. contains a nonzero divisor we have that T Ž R. s T Ž R. . U Ž Ž A . . s U Ž A . .T by Žw5, Lemma 1.1x. the system Ž Ž S .. has a solution. Žii. Let f be the canonical ring homomorphism from R to T Ž Rw T x.. Consider the T Ž Rw T x.-system
Ž f Ž S . . : f Ž A. . x s f Ž b . . Let u 0 , . . . , u h 4 be a system of generators of the ideal Ur Ž A.. Since Ur Ž A. is faithful we have that the element u 0 q u1.T q ⭈⭈⭈ qu h .T h is a nonzero divisor in Rw T x. Consequently T Ž Rw T x . s T Ž Rw T x . . U Ž f Ž A . . s U Ž A . .T Again by Žw5, Lemma 1.1x. the system Ž f Ž S .. has a solution. Let Ž S . : A. x s b be an overdetermined system. For each sequence i1 , . . . , i n4 with 1 F i1 - ⭈⭈⭈ - i n F m we consider the Ž n = n.-system
Ž Si , . . . , i . : A i , . . . , i 1
n
1
n
x s bi1 , . . . , i n ,
where a i11 .. A i1 , . . . , i n s . a i n1
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . , ai n n
0
and bi1 , . . . , i n s Ž bi1, . . . , bi n . t. If AUi1 , . . . , i n is the cofactors matrix of A i1 , . . . , i n, then AUi1 , . . . , i n . A i1 , . . . , i n . x s AUi1 , . . . , i n .bi1 , . . . , i n ,
632
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
or equivalently ⌬ i1 , i 2 , . . . , i n . x 1 s ⌬1i1 , i 2 , . . . , i nŽ A N b . ⌬ i1 , i 2 , . . . , i n . x 2 s ⌬2i1 , i 2 , . . . , i nŽ A N b . ⌬ i1 , i 2 , . . . , i n . x n s ⌬ni1 , i 2 , . . . , i nŽ A N b . , where ⌬ i1 , . . . , i n is the determinant of the matrix A i1 , . . . , i n and ⌬ ij1 , i 2 , . . . , i nŽ A N b . is the determinant
⌬ ij1 , i 2 , . . . , i nŽ A N b . s
a i11
⭈⭈⭈
a i1 jy1
bi1
a i1 jq1
⭈⭈⭈
ai1 n
ai 2 1 .. . a i n1
⭈⭈⭈
a i 2 jy1 .. . a i n jy1
bi 2 .. . bi n
a i 2 jq1 .. . a i n jq1
⭈⭈⭈
ai 2 n .. . . ai n n
⭈⭈⭈
⭈⭈⭈
For j s 1, 2, . . . , n consider the proportionality with Ž mn . equalities given by
Ž Pj . : ⌬ i , i 1
Ž A . x j s ⌬ ij1 , i 2 , . . . , i nŽ A N b . 4 1Fi1-i 2- ⭈⭈⭈ -i nFm .
2 , . . . , in
We put
Ž Pj . : ⌬ Ž A . x j s ⌬ j Ž A N b . , where ⌬Ž A. is the column vector Ž ⌬ i1 , i 2 , . . . , i nŽ A..1t F i1 - i 2 - ⭈⭈⭈ - i n F m and ⌬ j Ž A N b . the column vector Ž ⌬ ij1 , i 2 , . . . , i nŽ A N b ..1t F i1 - i 2 - ⭈⭈⭈ - i n F m for j s 1, 2, . . . , n. LEMMA 1.3. With the abo¨ e notations we ha¨ e: Ži.
If rankŽ A. s rankŽ A N b . s n then rank Ž ⌬ Ž A . . s rank Ž ⌬ Ž A . < ⌬ j Ž A N b . . s 1,
for j s 1, 2, . . . , n. Žii. If UnŽ A. s UnŽ A < b . then Un Ž A . s U1 Ž ⌬ Ž A . . s U1 Ž ⌬ Ž A . < ⌬ j Ž A N b . . . Proof. We can suppose that j s 1. We prove the equality ⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b . s ⌬ l 1 , . . . , l nŽ A . .⌬1i1 , . . . , i nŽ A N b . ,
633
LINEAR EQUATIONS
for 1 F i1 - i 2 - ⭈⭈⭈ - i n F m and 1 F l 1 - l 2 - ⭈⭈⭈ - l n F m. We have n
k
⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b . s⌬ i1 , . . . , i nŽ A . . Ý Ž y1 . .bl k .⌬2,l 1 ,. .. .. ., ,nl k , . . . , l n Ž A . , ks1
Ž 1. where
⌬2,l 1 ,. .. .. ., ,nl k , . . . , l n Ž A . s
al1 2 .. . a l ky 1 2
al1 3 .. . a l ky 1 3
⭈⭈⭈
a l kq 1 2 .. . al n 2
a l kq 1 3 .. . al n 3
⭈⭈⭈
⭈⭈⭈
⭈⭈⭈
al1 n .. . a l ky 1 n a l kq 1 n .. . al n n
.
Since Unq 1Ž A N b . s Ž0. it follows that bl k
al k 1
al k 2
⭈⭈⭈
al k n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
bi 2 .. . bi n
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n s 0, .. . ai n n
or equivalently bl k
al k 1
al k 2
⭈⭈⭈
al k n
0
al k 1
al k 2
⭈⭈⭈
al k n
0
a i11
ai1 2
⭈⭈⭈
ai1 n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
0 .. . 0
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n q bi 2 .. .. . . ai n n bi n
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n s 0. .. . ai n n
Hence 0
al k 1
al k 2
⭈⭈⭈
al k n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
bl k⌬ i1 , . . . , i nŽ A . s y bi 2 .. .
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
ai 2 n . .. . ai n n
bi n
634
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Substituting in Ž1. we obtain the chain of equalities ⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b .
n
sy
Ý Žy1. k ⌬2l , ,. .. .. ., ,nl , 1
k . . . , ln
ks1
0
a l k1
⭈⭈⭈
al k n
bi1
a i11 .. . a i n1
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . ai n n
Ž A.
⭈⭈⭈
Ž A. . .
. bi n
n
0
n
Ý Žy1. k al 1 ⌬2l , ,. .. .. ., ,nl , k
k . . . , ln
1
ks1
⭈⭈⭈ .. . ⭈⭈⭈
a i11 .. . a i n1
bi n ⌬ l 1 ⭈ ⭈ ⭈ l nŽ A .
0
⭈⭈⭈
0
a i11 .. . a i n1
ai1 2 .. . ai n2
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . ai n n
bi1 sy . .. bi n
k
k . . . , ln
1
Ž A.
ks1
s y bi1 .. .
0
Ý Žy1. k al n ⌬2l , ,. .. .. ., ,nl , ai1 n .. . ai n n
s ⌬ l 1 , . . . , l nŽ A . .⌬1i1 , . . . , i nŽ A N b . .
Note that for t ) 1 we have
n
Ý Ž y1. k al t ⌬ l ,2,. ....,.l, n, k
1
ks1
k
. . . , ln
al1 t
al1 2
⭈⭈⭈
al1 t
⭈⭈⭈
al1 n
al
al 2 2 .. . al n 2
⭈⭈⭈
al 2 t .. . al n t
⭈⭈⭈
al 2 n .. s 0. . al n n
Ž A . s .2 .
t
. al n t
⭈⭈⭈
⭈⭈⭈
Žii. It is clear. PROPOSITION 1.4. Let Ž S . : Ax s b be an o¨ erdetermined system where A is a Ž m = n.-matrix. Suppose that UnŽ A. is a faithful ideal. Then the following statements are equi¨ alent: Ži. Ž S . has a solution Ž necessarily unique. in R. Žii. For j s 1, 2, . . . , n the proportionality
Ž Pj . : ⌬ Ž A . . x j s ⌬ j Ž A N b . has a solution Ž necessarily unique. in R.
635
LINEAR EQUATIONS
In fact ␣ s Ž ␣ 1 , . . . , ␣ n . t is the unique solution of Ž S . if and only if ␣ j is the unique solution of Ž Pj . for j s 1, . . . , n. Proof. Assume Ži.. Suppose that ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ n . t is the solution of Ž S .. For each sequence i1 , i 2 , . . . , i n4 with 1 F i1 - i 2 - ⭈⭈⭈ - i n F m we have that ␣ is solution of the system Ž Si1 , . . . , i n .. So we have AUi1 , . . . , i n . A i1 , . . . , i n . ␣ s AUi1 , . . . , i n .bi1 , . . . , i n hence ⌬ i1 , . . . , i nŽ A . . ␣ j s ⌬ ij1 , . . . , i nŽ A N b . , for j s 1, 2, . . . , n. Consequently ␣ j is the solution of the system Ž Pj .. Assume Žii.. Let ␣ j be the solution of Ž Pj . for j s 1, 2, . . . , n. Consider the element ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ n . t. Since UnŽ A. is a faithful ideal of R then the element
s
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
Ý
ny 1
in
1Fi 1-i 2- ⭈⭈⭈ -i n Fm
⌬ i1 , . . . , i nŽ A .
is a nonzero divisor in the polynomial ring Rw Y x. For k s 1, 2, . . . , m we have in Rw Y x the following chain of equalities:
Ž a k1 ␣ 1 q ⭈⭈⭈ qak n ␣ n . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= ⌬ i1 , . . . , i nŽ A . . Ž a k1 ␣ 1 q ⭈⭈⭈ qa k n ␣ n . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ␣ 1 ⌬ i1 , . . . , i nŽ A . q ⭈⭈⭈ qa k n ␣ n ⌬ i1 , . . . , i nŽ A . . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . . .
636
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Now, since rankŽ A < b . s n it follows that a k1 a i11
bk bi1
ak 2 ai1 2
⭈⭈⭈ ⭈⭈⭈
ak n ai1 n
0 s ai 2 1 .. .
bi 2 .. . bi n
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
ai 2 n .. . ai n n
a i n1
s a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . y bk ⌬ i1 , . . . , i nŽ A . . Therefore
Ž a k1 ␣ 1 q ⭈⭈⭈ qak n ␣ n . s
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
Ý
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
s bk
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
bk ⌬ i1 ⭈ ⭈ ⭈ i nŽ A .
ny 1
1Fi 1- ⭈⭈⭈ -i n Fm
in
⌬ i1 ⭈ ⭈ ⭈ i nŽ A . s bk .
Since is a nonzero divisor it follows that a k1 ␣ 1 q ⭈⭈⭈ qa k n ␣ n s bk , and consequently ␣ is a solution of Ž S ..
2. CHARACTERIZATION THEOREMS THEOREM 2.1. Let R be a commutati¨ e ring with unit. Then the following statements are equi¨ alent: Ži. R is an integrally closed ring. Žii. A proportionality Ž P . : a. x s b such that U1Ž a. contains a nonzero di¨ isor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. An o¨ erdetermined system Ž S . : Ax s b, where A is an Ž m = n.matrix such that UnŽ A. contains a nonzero di¨ isor, has a solution in R if and only if Up Ž A. s Up Ž A < b . for p s n, n q 1.
637
LINEAR EQUATIONS
Proof. First we prove that Ži. and Žii. are equivalent. Assume Ži.. Let Ž P . : a. x s b be a proportionality such that U1Ž a. contains a nonzero divisor and suppose that rankŽ a. s rankŽ a < b . and U1Ž a. s U1Ž a < b .; we show that Ž P . has a solution in R. By Lemma 1.2, the proportionality Ž P . has a solution rs in the total quotient ring T Ž R ., because rankŽ a. s rankŽ a < b . s 1. Condition U1Ž a. s U1Ž a < b . assures that there exist elements c i j g R, 1 F i, j F n, such that bi s Ý njs1 c i j a j . Consequently in T Ž R . we have the equalities: a1 . a2 .
s
s
s c11 a1 q c12 a2 q ⭈⭈⭈ qc1 n a n s c 21 a1 q c 22 a2 q ⭈⭈⭈ qc2 n a n .. .
an .
s
s c n1 a1 q c n2 a2 q ⭈⭈⭈ qc n n a n .
If C is the Ž n = n.-matrix C s Ž c i j ., then the above equalities can be rewritten in the form
ž
s
.Id n y C a s 0,
/
where Id n is the identity matrix of order n. Therefore det
ž
s
.Id n y C .a i s 0,
/
for i s 1, 2, . . . , . Since U1Ž a. contains a nonzero divisor it follows that 0 s det
ž
s
.Id n y C s
n
/ ž / s
q c ny1
ny1
q ⭈⭈⭈ qc1
ž / s
ž / s
q c0 ,
where c i g R for i s 0, 1, . . . , n y 1. Thus s is integer over R and hence g R because R is an integrally closed ring. s Assume Žii.. Let rs be an element of T Ž R . such that
ž / s
n
q c ny 1
ž / s
ny1
q ⭈⭈⭈ qc1
ž / s
q c 0 s 0,
where c i g R for i s 0, 1, . . . , n y 1. Consider the system of linear equa-
638
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
tions Ž P . : ax s b given by
¡s
Ž P . :~
x x
s s
ny 2 sx ny 1 x
s s
ny 1
s
ny 2
.. .
¢
s ny2 2 s ny3 .. . y Ž c0 s
ny1
ny1 q c1 s ny2 q ⭈⭈⭈ qc ny1 ny1 . .
For Ž P . one has rank Ž a . s rank Ž a < b . s 1, U1 Ž a . s U1 Ž a < b . s Ž s ny 1 , s ny2 , . . . , ny2 s, ny1 . . Moreover U1Ž a. contains the nonzero divisor s ny 1. But Žii. the proportionality Ž P . has a unique solution in R that obviously is also the unique solution in T Ž R .. Since is the solution of Ž P . in T Ž R . it follows that s
g R. Next we prove that Žii. and Žiii. are equivalent. Clearly Žii. implies Žiii.. Conversely, let A be an Ž m = n.-matrix such that UnŽ A. contains a nonzero divisor. Suppose that the overdetermined system Ž S . : Ax s b satisfies Up Ž A. s Up Ž A N b . for p s n, n q 1 Ži.e., rankŽ A. s rankŽ A < b . s n and UnŽ A. s UnŽ A N b .. Consider the proportionalities s
Ž Pj . : ⌬ Ž A . x j s ⌬ j Ž A N b . ,
j s 1, 2, . . . , n
introduced in Section 1. By Lemma 1.3, we have rank Ž ⌬ Ž A . . s rank Ž ⌬ Ž A . < ⌬ j Ž A N b . . s 1, and U 1 Ž ⌬ Ž A . . s U1 Ž ⌬ Ž A . < ⌬ j Ž A N b . .
for j s 1, 2, . . . , n.
Moreover U1Ž ⌬Ž A.. s UnŽ A. contains a nonzero divisor. By Žiii. the proportionality Ž Pj . for j s 1, 2, . . . , n has a solution in R and hence, by Proposition 1.4, the system Ž S . has a solution in R. COROLLARY 2.2. are equi¨ alent:
Let R be an integral domain. The following statements
Ži. R is an integrally closed domain. Žii. A proportionality Ž P . : ax s b has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. An o¨ erdetermined system Ž S . : Ax s b, where A is an Ž m = n.matrix has a solution in R if and only if Up Ž A. s Up Ž A N b . for p s n, n q 1.
639
LINEAR EQUATIONS
THEOREM 2.3. equi¨ alent:
Let R be a commutati¨ e ring. The following statements are
Ži. The local ring R ᒊ is an integrally closed domain for e¨ ery maximal ᒊ of R. Žii. A proportionality Ž P . : ax s b has a solution in R if and only if Ž rank a. s rankŽ a < b . and Up Ž a. s Up Ž a < b . for p s 1, 2. Proof. Assume Ži.. Let Ž P . : ax s b be a proportionality such that Up Ž a. s Up Ž a < b . for p s 1, 2. By Žw5, Proposition 1x. the proportionality Ž P . has a solution in R if and only if Ž Pᒊ . has a solution in R ᒊ for every maximal ideal ᒊ of R. Let ᒊ be a maximal ideal of R. If U1Ž a.. R ᒊ s U1Ž a < b .. R ᒊ s 0 then Ž Pᒊ . is the trivial system of linear equations Ž Pᒊ . : 0. x s 0 and hence Ž Pᒊ . has a solution in R ᒊ . On the other hand if U1Ž a.. R ᒊ s U1Ž a < b .. R ᒊ / 0 then the proportionality
Ž Pᒊ . :
a 1
.x s
b 1
satisfies a
Up
ž / 1
s Up Ž a . . R ᒊ s Up Ž a < b . . R ᒊ s Up
a b
ž / 1 1
for p s 1, 2.
By Corollary 2.2 Ž Pᒊ . has a solution in R ᒊ . Assume Žii.. Let ᒊ be a maximal ideal of R. First we prove that R ᒊ is a u ¨ domain. Suppose that there are two elements u, ¨ in R such that 1 , 1 are u
¨
nonzero elements in R ᒊ and 1 . 1 s 0 or equivalently s.u.¨ s 0 for some s f ᒊ. Then the proportionality Ž P . : a. x s b given by xs0 Ž P . : s.u. ¨ . x s s.¨
½
satisfies rankŽ a. s rankŽ a < b . s 1 and U1Ž a. s U1Ž a < b . s Ž u, s.¨ .. However Ž P . has no solution in R because Ž Pᒊ . has no solution in R ᒊ , see Žw3, Theorem 6x.. Next we prove that R ᒊ is an integrally closed domain. Let Ž P˜. : ax ˜ s ˜b by a proportionality in R ᒊ given by
Ž P˜. :
½
ai si
.x s
bi ti
5
, 1FiFm
such that Up Ž a ˜. s Up Ž a˜< ˜b . for p s 1, 2. We show that Ž P˜. has a solution in R ᒊ .
640
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Since U2 Ž a ˜< ˜b . s Ž0. there exists s f ᒊ such that s Ž t i .a i .s j .bj y si .bi .t j .a j . s 0 for 1 F i , j F m. Moreover there exists t f ᒊ such that t.si .bi g Ž t 1 .a1 , t 2 .a2 , . . . , t m .a m . , because U1Ž a ˜. s U1Ž a˜< ˜b .. It follows that the proportionality Ž P . : ax s b over R given by
Ž P . : t i .ai . x s s.t.si .bi 4 1FiFm satisfies Up Ž a. s Up Ž a < b . for p s 1, 2. By Ži. Ž P . has a solution ␣ in R and ␣ hence s.t is the solution of Ž P˜.. 3. WHEN IS THE POLYNOMIAL RING Rw T x INTEGRALLY CLOSED? PROPOSITION 3.1. grally closed. Then Ži. Žii. ideal has Žiii.
Let R be a commutati¨ e ring such that Rw T x is inte-
R is a reduced ring. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. R is integrally closed.
Proof. Ži. If a is a nonzero nilpotent element of R then
a T
is an
integral over R and however T f Rw T x. Žii. Since U1Ž a. is a faithful ideal of R it follows that U1Ž a. Rw T x contains a nonzero divisor in Rw T x. By Theorem 2.1, Ž P . has a solution in Rw T x. Therefore Ž P . has a solution in R because the extension R ª Rw T x is faithfully flat. Žiii. This is a straightforward application of Theorem 2.1 and the above statement. a
The main result of this section establishes that Ži. and Žii. are sufficient conditions to assure that Rw T x is integrally closed. Before starting the main result of this section we need some previous results:
641
LINEAR EQUATIONS
LEMMA 3.2. Let R be a reduced ring and Ž P . : f. x s g be a proportionality in Rw T x gi¨ en by
¡ f ŽT . . x s g ŽT . 1
Ž P . :~
¢f
1
f2 Ž T . . x s g2 Ž T . .. . m
Ž T . . x s gm Ž T . .
Suppose that U1Ž f . is a faithful ideal. If ␣ ŽT . is a solution of Ž P . then
⭸ ⬚␣ Ž T . F max ⭸ ⬚g 1 Ž T . , ⭸ ⬚g 2 Ž T . , . . . , ⭸ ⬚g m Ž T . 4 , where ⭸ ⬚␣ ŽT . denotes the degree of the polynomial ␣ ŽT .. Proof. Put f i Ž T . s a i0 q a i1T q ⭈⭈⭈ qa i p i T p i , g i Ž T . s bi0 q bi1T q ⭈⭈⭈ qbi q i T q i , for i s 1, 2, . . . , m. Let ␣ ŽT . s ␣ 0 q ␣ 1T q ⭈⭈⭈ q␣ r T r be the solution of Ž P .. Suppose that r ) qi for i s 1, 2, . . . , m. Then we have the equalities ai p i ␣ r s 0 a i p iy 1 ␣ r q a i p i ␣ ry1 s 0 .. . a i0 ␣ r q a i1 ␣ ry1 q ⭈⭈⭈ s 0, for i s 1, 2, . . . , m. It follows that a i p i ␣ rsa i p iy1 ␣ r2s ⭈⭈⭈ sa i0 ␣ rrq1 s 0,
1FiFm.
f i ŽT .. ␣ rrq1
Hence s 0 for i s 1, . . . , m and consequently U1Ž f .. ␣ rrq1 s 0. Since R is reduced we have that ␣ rrq1 / 0. This provides the contradiction that we were seeking because, by hypothesis, U Ž f . is faithful ideal. Let f ŽT . s a0 q a1T q ⭈⭈⭈ qa p T p be an element of Rw T x. We denote ᑾ f the content of f ŽT ., Ži.e., the ideal of R generated by the coefficients of f ŽT ... For each positive integer s we denote by AŽ f ; s . the ŽŽ p q s . = s .matrix
¡a
¦
a1 a2 A Ž f ; s . s .. .
0 a0 a1 .. .
0 0 a0 .. .
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
0 0 0 .. .
0 0 0 .. .
0
0
0
⭈⭈⭈
ap
a py1
0
0
⭈⭈⭈
0
ap
0
¢0
.
§
642
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
LEMMA 3.3.
For a positi¨ e integer r with 1 F r F s we ha¨ e Ur Ž A Ž f ; s . . s ᑾ rf .
Proof. We use induction on r. When r s 1 we have the equality U1 Ž A Ž f ; s . . s ᑾ f , for each positive integer s. We now suppose that r ) 1 and that Ury1 Ž A Ž f ; s . . s ᑾ ry1 , f for each positive integer s such that r y 1 F s. Clearly Ur Ž AŽ f ; s .. is contained in ᑾ rf . By the induction hypothesis we have Ury1 Ž A Ž f ; s . . s ᑾ f .U Ury1 Ž A Ž f ; s y 1 . . . s ᑾ f .U ᑾ rf s ᑾ f .ᑾ ry1 f Next we prove, by induction of k, that a k Ury1 Ž A Ž f ; s y 1 . . : Ur Ž A Ž f ; s . . , for k s 0, 1, . . . , n. Consider the matrix AŽ f, s y 1. as a submatrix of AŽ f, s . by
¡a
0
0
⭈⭈⭈
¦
0
a1 .. .
AŽ f ; s . s a p 0 .. .
¢0
.
AŽ f ; s y 1.
§
So it is clear that a0 Ury1 Ž A Ž f ; s y 1 . . : Ur Ž A Ž f ; s . . . Let k ) 0 and suppose that a s Ury1 Ž A Ž f ; s y 1 . . s a s Ury1 Ž A Ž f ; s . . : Ur Ž A Ž f ; s . . , ry 1 be the submatrix of for all s F k y 1. Let AŽ f, s y 1. ij11,, .. .. .. ,, ijry AŽ f, s . 1 composed by the rows i1 , i 2 , . . . , i ry1 with 2 F i1 - i 2 - ⭈⭈⭈ - i ry1 F p q s and the columns j1 , j2 , . . . , jry1 with 2 F j1 - j2 - ⭈⭈⭈ - jry1 F s. We con-
643
LINEAR EQUATIONS
sider the matrix
¡a
a kyj1q1
k
¦
⭈⭈⭈
a kyj 2q1
a i1q1 B s a i q1 2 .. . .. .
j ,..., j
ry 1 A Ž f ; s y 1 . i11, . . . , i ry 1
¢
,
§
where for a positive integer i we put ayi s a pqi s 0. Note that if k q 1 is j1 , . . . , j ry 1 different of i1 , i 2 , . . . , i ry1 then B is the submatrix of AŽ f ; s .1,kq1, i 1 , . . . , i ry 1 . If k q 1 is equal to i s for some s then the determinant of B is zero and hence detŽ B . s 0 g Ur Ž AŽ f ; s ... Therefore ry1
j , . . . , ju
Ý akyj q1 . ␦t g Ur Ž AŽ f ; s . . ,
det Ž B . s a k .det A Ž f ; s y 1 . i11, . . . , i ryry1 1 q
ž
/
t
ts1
where ␦ t is the adjoint on B associated to the element a kyj tq1 of the first row. Consequently ␦ t g Ury1Ž AŽ f ; s .. and hence j ,..., j
ry 1 det Ž B . s a k det A Ž f ; s y 1 . i11, . . . , i ry 1
ž
/
Ury1 Ž A Ž f ; s . . . q terms in Ž a0 , a1 , . . . , a ky1 . .U Since detŽ B . g Ur Ž AŽ f ; s .. and, by the induction hypothesis,
Ž a0 , . . . , ary1 . Ury1 Ž A Ž f ; s . . : Ur Ž A Ž f ; s . . , it follows that j ,..., j
ry 1 a k .det A Ž f ; s y 1 . i11, . . . , i ry g Ur Ž A Ž f ; s . . . 1
ž
/
COROLLARY 3.4. Let f 1ŽT ., f 2 ŽT ., . . . , f mŽT . be elements of Rw T x and let s be a positi¨ e integer. If AŽ f 1 , f 2 , . . . , f m ; s . is the block matrix AŽ f 1 ; s . AŽ f 1 , f 2 , . . . , fm ; s . s
0
AŽ f 2 ; s . , .. .
AŽ fm ; s .
644
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
then Ur Ž A Ž f 1 , f 2 , . . . , f m ; s . . s
Ý
r1qr 2q ⭈⭈⭈ qr m sr
ᑾ rf 11 ᑾ rf 22 ⭈⭈⭈ ᑾ rf mm ,
for 1 F r F s. Proof. Let pi be the degree of f i for i s 1, 2, . . . , m. For a positive integer s we have that AŽ f 1 , f 2 , . . . , fm ; s . s AŽ f ; s . , where f ŽT . is the polynomial f Ž T . s f 1 q T p 1qs f 2 q T p 1qp 2q2 s f 3 q ⭈⭈⭈ qT p 1qp 2q ⭈⭈⭈ qp my 1qŽ my1. s f m , since
ᑾ f s ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m ,
the result follows from the above lemma. We denote by RŽT . the quotient ring Sy1 Rw T x of the polynomial ring Rw T x where S is the multiplicative closed set of f g Rw T x such that ᑾ f s R. LEMMA 3.5. Let R be a ring and let Y be an element of RŽT . that is integral o¨ er Rw T x. Then there exists h g Rw T x such that Y y h is nilpotent in RŽT .. Proof. See Žw6, Lemma 16.2x.. The main result of this section is: THEOREM 3.6. Let R be a commutati¨ e ring. Then the following statements are equi¨ alent: Ži. Rw T x is an integrally closed ring. Žii. R satisfies the following properties: ᎏ R is a reduced ring. ᎏ A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. R satisfies the following properties: ᎏ R is a reduced ring. ᎏ An o¨ erdetermined system Ž S . : A. x s b where A is an Ž m = n.-matrix such that UnŽ A. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s n, n q 1. Proof. Ži. implies Žii. by Proposition 3.1. The equivalence between Žii. and Žiii. follows from Proposition 1.3. We show that Žiii. implies Ži. using Theorem 2.1
645
LINEAR EQUATIONS
Suppose that
¡f ŽT . . xŽT . s g ŽT . 1
Ž P . :~
1
f2 Ž T . . x Ž T . s g2 Ž T . .. .
¢f
m
Ž T . . x Ž T . s gm Ž T .
is a proportionality, denoted by Ž P . : f. x s g, in Rw T x such that U1Ž f . contains a nonzero divisor in Rw T x and Up Ž f . s Up Ž f < g . for p s 1, 2. Let m
FŽT . s
Ý i Ž T . . f i Ž T . g U1 Ž f Ž T . . is1
be a nonzero divisor, and consider m
GŽ T . s
Ý i Ž T . . g i Ž T . . is1
GŽT . is the solution of Ž P . in T Ž Rw T x.. Moreover s F Ž T . is an integer over Rw T x, see Theorem 2.1 Put f i Ž T . s a i0 q a i1T q ⭈⭈⭈ qa i p i T p i ,
Then s
GŽ T . F ŽT .
g i Ž T . s bi0 q bi1T q ⭈⭈⭈ qbi q i T q i , for i s 1, 2, . . . , m and q s max q1 , q2 , . . . , qm 4 . By Lemma 3.2, if Ž P . has a solution x ŽT . Žnecessarily unique. in Rw T x then x ŽT . is the form x ŽT . s x 0 q x 1T q ⭈⭈⭈ qx q T q. Consequently the proportionality Ž P . has a solution in Rw T x if and only if the system of linear equations over R,
Ž S . : AŽ f 1 , f 2 , . . . , fm ; q q 1. x s b has a solution in R, where AŽ f 1 , f 2 , . . . , f m ; q q 1. is defined in Corollary 3.4, x s Ž x 0 , x 1 , . . . , x q . t is the column vector of indeterminates and b is the column vector b s Ž b10 , . . . , b1 q1 , 0, . . . , 0, b 20 , . . . , b 2 q 2 , t
0, . . . , 0, . . . , bm 0 , . . . , bm q m , 0, . . . , 0 . , Next we show that Ž S . satisfies the conditions of statement Žiii.. By Corollary 3.4, we have Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s
Ý
r1qr 2q ⭈⭈⭈ qr m sqq1
ᑾ rf 11 ᑾ rf 22 ⭈⭈⭈ ᑾ rf mm
s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
qq 1
.
646
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Since the ideal of Rw T x generated by f 1ŽT ., f 2 ŽT ., . . . , f mŽT . contains a nonzero-divisor it follows that ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m is a faithful ideal of R. The hypothesis U1Ž f . s U1Ž f < g . implies that ᑾ g i : ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m , for i s 1, 2, . . . , m. So we have Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . : Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . : Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . q Ž ᑾ g 1 q ᑾ g 2 q ⭈⭈⭈ qᑾ g m . Uq Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
qq 1
q Ž ᑾ g 1 q ᑾ g 2 q ⭈⭈⭈ qᑾ g m . . Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m . s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
q
qq 1
s Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . . Therefore Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s Uqq1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . . Finally we prove that Uqq 2 Ž AŽ f 1 , f 2 , . . . , f m ; q q 1.< b . s Ž0.. Consider the canonical inclusion of R in R⬘ s T Ž Rw Z x. where Z is a new indeterminate and note that the proportionality Ž P . has a solution in R⬘w T x if and only if the system Ž S . has a solution in R⬘. Next we show that Ž P . has a solution in R⬘w T x. Replacing T by T q Z, we obtain the proportionality
¡ f ŽT . . xŽT . s g ŽT .
Ž P⬘ . :~
¢f
X 1 f 2X
X m
X 1 g X2
ŽT . . xŽT . s ŽT . .. .
Ž T . . x Ž T . s g Xm Ž T . ,
where f iX ŽT . s f i ŽT q Z . and g Xi ŽT . s g i ŽT q Z . for i s 1, 2, . . . , m. The unique solution of Ž P⬘. in T Ž R⬘w T x. is ⬘ s G⬘ŽT .rF⬘ŽT . where F⬘ŽT . s F ŽT q Z . and G⬘ŽT . s GŽT q Z .. Moreover ⬘ is integral over R⬘w T x. Since F Ž Z . is a nonzero divisor on Rw Z x it follows that F⬘Ž0. s F Ž Z . is a
LINEAR EQUATIONS
647
unit in R⬘ and hence ⬘ s G⬘ŽT .rF⬘ŽT . g R⬘ŽT .. By Lemma 3.5 there exists h g R⬘w T x such that ⬘ y h is nilpotent in R⬘ŽT .. Consequently ⬘ s h g R⬘w T x because R is a reduced ring. Thus Ž P⬘. has a solution in R⬘w T x and therefore Ž P . has a solution in R⬘w T x. Since Ž P . has a solution in R⬘w T x it follows that Ž S . has a solution in R⬘. Consequently Uqq 2 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . . R⬘ s Ž 0 . , and hence Uqq 2 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . s Ž 0 . , because R is contained in R⬘. The system Ž S . satisfies the conditions of statement Žiii.. Thus Ž S . has a solution in R and hence has solution in Rw T x. Recall that the set Spec R of all prime ideals of R together with its Zariski topology is quasi-compact. However, the subset Min R of all minimal prime ideals of R in general is not quasi-compact. LEMMA 3.7. Let R be a reduced ring such that Min R is quasi-compact. If ᑾ is a finitely generated ideal of R then there exists a finitely generated ideal ᑿ : AnnŽ ᑾ . such that ᑾ q ᑿ is a faithful ideal. Proof. Let ᑾ be a finitely generated ideal of R. Since R is a reduced then Rw T x satisfies the annihilator condition, see Žw6, pp. 5 and 7x.. Hence there exists a polynomial g ŽT . g Rw T x such that Ann Ž ᑾ . R w T x . s Ann Ž g Ž T . . . The canonical homomorphism i: R ª Rw T x induces a homeomorphism between Min R and Min Rw T x. Consequently Min Rw T x is quasi-compact. By Žw6, Theorem 4.3x. there exist polynomials h1ŽT ., h 2 ŽT ., . . . , h k ŽT . in AnnŽ g ŽT .. such that Ann Ž h1 Ž T . , h 2 Ž T . , . . . , h k Ž T . , g Ž T . . s 0. Let ᑿ be the ideal of R generated by all coefficients of the polynomials h1ŽT ., h 2 ŽT ., . . . , h k ŽT .. Then we have that ᑿ : AnnŽ ᑾ ., because h i ŽT . : Ann Ž g Ž T .. s Ann Ž ᑾ R Ž T .. , and Ann Ž ᑾ q ᑿ . s Ž 0 . , because AnnŽ h1ŽT ., h 2 ŽT ., . . . , h k ŽT .. l AnnŽ ᑾ RŽT .. s Ž0.. COROLLARY 3.8. Let R be a reduced ring such that with Min R quasicompact. Then the following statements are equi¨ alent: Ži. A proportionality Ž P . : a. x s b o¨ er R has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2.
648 Žii. Žiii. Živ. ideal has
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
R ᒊ is an integrally closed domain for e¨ ery ᒊ maximal of R. Rw T x is an integrally closed ring. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2.
Proof. By Theorems 2.3 and 3.1 it is sufficient to prove that Živ. implies Ži.. Let Ž P . : a. x s b be a proportionality in R such that Up Ž a. s Up Ž a < b . for p s 1, 2. By Lemma 3.7 there exist c i g AnnŽ Ui Ž a.., i s 1, 2, . . . , t such that Ann Ž ² c1 , c 2 , . . . , c k , a1 , a2 , . . . , a n : . s 0. The proportionality
¡a . x s b 1
Ž P⬘ . :~
1
a2 . x s b 2 .. . a n . x s bn , c1 . x s 0 c2 . x s 0 .. .
¢c . x s 0 k
by Živ. has a solution in R and hence Ž P . has a solution in R. Recall that R satisfies Property A if each finitely generated ideal of R composed of zero divisors has nonzero annihilator. Equivalently R satisfies Property A if and only if each generated faithful ideal of R contains a nonzero divisor. COROLLARY 3.9. Let R be a reduced ring satisfying Property A. Then the following statements are equi¨ alent: Ži. Rw T x is an integrally closed ring. Žii. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. A proportionality Ž P . : a. x s b o¨ er R such that Ui Ž a. contains a nonzero di¨ isor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Živ. R is an integrally closed ring.
649
LINEAR EQUATIONS
REFERENCES 1. T. Akiba, Integrally-closedness of polynomial rings, J. Math. Japan 6 Ž1980., 67᎐75. 2. P. Camion, L. S. Levi, and H. B. Mann, Linear equations over a commutative ring, J. Algebra 19 Ž1972., 432᎐446. 3. J. A. Hermida-Alonso and T. Sanchez-Giralda, Linear equations over commutative rings ´ and determinantal ideals, J. Algebra 99, No. 1 Ž1986., 72᎐79. 4. J. A. Hermida-Alonso, Linear equations over commutative rings and grade theory, J. Algebra 148, No. 2 Ž1992., 497᎐503. 5. J. A. Hermida-Alonso, Linear equations over commutative rings and finite free resolutions, J. Algebra 164, No. 2 Ž1994., 452᎐467. 6. J. A. Huckaba, ‘‘Commutative Rings with Zero Divisors,’’ Dekker, New York, 1988. 7. T. Lucas, Characterizing when Rw X x is integrally closed, Proc. Amer. Math. Soc. 105 Ž1989., 861᎐867. 8. T. Lucas, Characterizing when Rw X x is integrally closed II, J. Pure Appl. Algebra 61 Ž1989., 49᎐52. 9. T. Lucas, The complete integral closure of Rw X x, Trans. Amer. Math. Soc. 330 Ž1992., 757᎐768. 1 10. N. H. McCoy, Rings and ideals, Carus Math. Monographs 8, No. Ž1948.. 4
11. D. G. Northcott, ‘‘Finite Free Resolutions,’’ Cambridge Univ. Press, Cambridge, U.K., 1976. 12. Y. Quentel, Sur la compacite ´ du spectre minimal d’un anneau, Bull. Soc. Math. France 99, No. 18 Ž1971., 265᎐272. 13. D. W. Sharpe, Grade and the theory of linear equations, Linear Algebra Appl. 18 Ž1977., 25᎐32. 14. W. V. Vasconcelos, Annihilators of modules with a finite free resolution, Proc. Amer. Math. Soc. 29 Ž1971., 440᎐442.
Characterizing when the Rings R and Rw T x Are Integrally Closed via Linear Equations over R A. DeFrancisco-Iribarren* and J. A. Hermida-Alonso† Departamento de Matematicas, Uni¨ ersidad de Leon, ´ ´ 24071 Leon, ´ Spain Communicated by D. A. Buchsbaum Received July 17, 1998
INTRODUCTION Let R be a commutative ring with an identity element. This paper is devoted to study when R and the polynomial ring Rw T x are integrally closed in terms of the systems of linear equations over R. Denote by T Ž R . the total quotient ring of R Ži.e., the ring of fractions of R with respect to the multiplicative set of all nonzero divisors of R .. The ring R is integrally closed if every x g T Ž R . which is integral over R belongs to R. If R is an integrally closed domain then so is the polynomial ring an one indeterminate Rw T x. On the other hand if R is an integrally closed ring with nonzero nilradical then Rw T x is never integrally closed Žthe element arT is integral over, but does not belong to, Rw T x for every nonzero nilpotent a in R .. The question is: When is the polynomial ring Rw T x integrally closed? Necessary conditions for Rw T x to be integrally closed are that R is reduced and integrally closed. However these conditions are not sufficient, see Žw6, p. 187x.. T. Akiba w1x proves that if R is a reduced ring such that for each maximal ideal ᒊ of R the local ring R ᒊ is an integrally closed domain then Rw T x is integrally closed. In the same paper it is proven that if R is a reduced integrally closed with Min R compact then Rw T x is integrally closed if and only if T Ž R . is a Von Neumann regular ring. T. Lucas gave necessary and sufficient conditions in order that the polynomial ring Rw T x be integrally closed. In w7x it is shown that Rw T x is integrally closed if and only if R is integrally closed in T Ž Rw T x.. In w8x it is *, † Partially supported by DGICYT PB95-0603-C02 and Junta Castilla y Leon ´ LE09r95. 628 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
LINEAR EQUATIONS
629
shown that Rw T x is integrally closed if and only if R is integrally closed in the ring of finite fractions of R. Let Ž S . : A. x s b a system of linear equations over R. If Ž S . has a solution in R then Up Ž A . s Up Ž A < b . for p G 0, where Up Ž A. denotes the pth-determinantal ideal of A and Ž A < b . denotes the augmented matrix of Ž S .. In general, the converse is not true. In w3x it is proven that the converse is true for any system of linear equations over R if and only if R is a Prufer ¨ ring. Next we expose the main results of this paper. The system Ž P . : a. x s b given by a1 . x s b1 a2 . x s b 2 Ž P. : .. .
¡
~
¢a
m.x
s bm
is called a proportionality. In Section 2 we prove that R is an integrally closed ring if and only if any proportionality Ž P . : a. x s b such that U1Ž a. contains a nonzero divisor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. We also characterize the rings R such that R ᒊ is an integrally closed domain for each maximal ideal ᒊ of R in terms of the all proportionalities over R. Section 3 is devoted to solve the question, when is Rw T x an integrally closed ring? In fact we prove that Rw T x is inegrally closed if and only if R is reduced and any proportionality Ž P . : a. x s b such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. As a consequence of the above characterizations we prove that if R is a reduced ring satisfying Property A then Rw T x is integrally closed if and only if R is integrally closed. Akiba w1x established this result using a different technique. Finally we prove that if Min R is quasi-compact then Rw T x is integrally closed if and only if R ᒊ is a domain integrally closed for every maximal ideal ᒊ of R. 1. OVERDETERMINED SYSTEMS OF LINEAR EQUATIONS Let Ž S . : A. x s b be a system of linear equations where A s Ž a i j . is a Ž m = n.-matrix with entries in R, b s Ž b1 , . . . , bm . t is a column vector with entries in R and x s Ž x 1 , . . . , x n . t is the column of indeterminates.
630
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
If f : R ª R⬘ is a ring homomorphism we denote by Ž f Ž S .. the R⬘system given by Ž f Ž S .. : f Ž A.. x s f Ž b . where f Ž A. s Ž f Ž a i j .. and f Ž b . s Ž f Ž b1 ., . . . , f Ž bm .. t. If R is contained on R⬘ and f is the inclusion homomorphism we say that Ž S . has a solution in R⬘ when the R⬘-system Ž f Ž S .. has a solution Žin R⬘.. We denote by rank R Ž A. the rank of A; i.e., the largest nonnegative integer p such that Up Ž A. / 0 where Up Ž A. is the ideal of R generated by all p = p-minors of A. If Ž S . : A. x s b has a solution in R then we have Up Ž A . s Up Ž A N b .
for p G 0,
where Ž A N b . is the augmented matrix of the system Ži.e., the Ž m = Ž n q 1..-matrix obtained from A by adding the column matrix b .. We say that Ž S . : A. x s b, where A is an Ž m = n.-matrix, is an overdetermined system if rank R Ž A. s n. The particular case
¡a . x s b 1
1
~
a2 . x s b 2 Ž P. : .. .
¢a
m.x
s bm
is called a proportionality and it is denoted by Ž P . : a. x s b. The next result is known as McCoy’s theorem, see w10, 11, p. 63x. THEOREM 1.1. The o¨ erdetermined homogeneous system Ž Sh . : A. x s 0 where A is a Ž m = n.-matrix uniquely has the tri¨ ial solution if and only if UnŽ A. is a faithful ideal Ž i.e., Ž0 : UnŽ A.. s 0.. Remark. Let J be an ideal of R. If R is a Noetherian ring then J is a faithful ideal if and only if grR J G 1, where grR J is the classical grade of J on R; i.e., the upper bound of the lengths of all R-sequences of elements of J. W. V. Vasconcelos in w14x constructs a faithful finitely generated ideal J in a commutative ring R such that every element of J is a zero divisor in R. The true grade or polynomial grade of J on R, notion due to M. Hochster, is defined by Gr R J s lim grRwT1 , . . . , T n x J . R w T1 , . . . , Tn x . nª⬁
Note that Gr R J G 1 if and only if J is a faithful ideal and when R is a Noetherian ring Gr R J s grR J. D. W. Sharpe w13x and J. A. Hermida-Alonso w4, 5x give sufficient conditions involving the true grade of the ideals Up Ž A. to assure that Ž S . has a solution in R.
631
LINEAR EQUATIONS
LEMMA 1.2. Let Ž S . : A. x s b an o¨ erdetermined system such that rank R Ž A. s rank R Ž A N b . s r. Then Ži. If Ur Ž A. contains a nonzero di¨ isor then Ž S . has a solution Ž necessarily unique. in the total quotient ring T Ž R . of R. Žii. If Ur Ž A. is a faithful ideal then Ž S . has a solution Ž necessarily unique. in the total quotient ring T Ž Rw T x. of Rw T x. Proof. Ži. Let be the canonical ring homomorphism from R to T Ž R .. Consider the T Ž R .-system
Ž Ž S . . : Ž A. . x s Ž b . . Since U Ž A. contains a nonzero divisor we have that T Ž R. s T Ž R. . U Ž Ž A . . s U Ž A . .T by Žw5, Lemma 1.1x. the system Ž Ž S .. has a solution. Žii. Let f be the canonical ring homomorphism from R to T Ž Rw T x.. Consider the T Ž Rw T x.-system
Ž f Ž S . . : f Ž A. . x s f Ž b . . Let u 0 , . . . , u h 4 be a system of generators of the ideal Ur Ž A.. Since Ur Ž A. is faithful we have that the element u 0 q u1.T q ⭈⭈⭈ qu h .T h is a nonzero divisor in Rw T x. Consequently T Ž Rw T x . s T Ž Rw T x . . U Ž f Ž A . . s U Ž A . .T Again by Žw5, Lemma 1.1x. the system Ž f Ž S .. has a solution. Let Ž S . : A. x s b be an overdetermined system. For each sequence i1 , . . . , i n4 with 1 F i1 - ⭈⭈⭈ - i n F m we consider the Ž n = n.-system
Ž Si , . . . , i . : A i , . . . , i 1
n
1
n
x s bi1 , . . . , i n ,
where a i11 .. A i1 , . . . , i n s . a i n1
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . , ai n n
0
and bi1 , . . . , i n s Ž bi1, . . . , bi n . t. If AUi1 , . . . , i n is the cofactors matrix of A i1 , . . . , i n, then AUi1 , . . . , i n . A i1 , . . . , i n . x s AUi1 , . . . , i n .bi1 , . . . , i n ,
632
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
or equivalently ⌬ i1 , i 2 , . . . , i n . x 1 s ⌬1i1 , i 2 , . . . , i nŽ A N b . ⌬ i1 , i 2 , . . . , i n . x 2 s ⌬2i1 , i 2 , . . . , i nŽ A N b . ⌬ i1 , i 2 , . . . , i n . x n s ⌬ni1 , i 2 , . . . , i nŽ A N b . , where ⌬ i1 , . . . , i n is the determinant of the matrix A i1 , . . . , i n and ⌬ ij1 , i 2 , . . . , i nŽ A N b . is the determinant
⌬ ij1 , i 2 , . . . , i nŽ A N b . s
a i11
⭈⭈⭈
a i1 jy1
bi1
a i1 jq1
⭈⭈⭈
ai1 n
ai 2 1 .. . a i n1
⭈⭈⭈
a i 2 jy1 .. . a i n jy1
bi 2 .. . bi n
a i 2 jq1 .. . a i n jq1
⭈⭈⭈
ai 2 n .. . . ai n n
⭈⭈⭈
⭈⭈⭈
For j s 1, 2, . . . , n consider the proportionality with Ž mn . equalities given by
Ž Pj . : ⌬ i , i 1
Ž A . x j s ⌬ ij1 , i 2 , . . . , i nŽ A N b . 4 1Fi1-i 2- ⭈⭈⭈ -i nFm .
2 , . . . , in
We put
Ž Pj . : ⌬ Ž A . x j s ⌬ j Ž A N b . , where ⌬Ž A. is the column vector Ž ⌬ i1 , i 2 , . . . , i nŽ A..1t F i1 - i 2 - ⭈⭈⭈ - i n F m and ⌬ j Ž A N b . the column vector Ž ⌬ ij1 , i 2 , . . . , i nŽ A N b ..1t F i1 - i 2 - ⭈⭈⭈ - i n F m for j s 1, 2, . . . , n. LEMMA 1.3. With the abo¨ e notations we ha¨ e: Ži.
If rankŽ A. s rankŽ A N b . s n then rank Ž ⌬ Ž A . . s rank Ž ⌬ Ž A . < ⌬ j Ž A N b . . s 1,
for j s 1, 2, . . . , n. Žii. If UnŽ A. s UnŽ A < b . then Un Ž A . s U1 Ž ⌬ Ž A . . s U1 Ž ⌬ Ž A . < ⌬ j Ž A N b . . . Proof. We can suppose that j s 1. We prove the equality ⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b . s ⌬ l 1 , . . . , l nŽ A . .⌬1i1 , . . . , i nŽ A N b . ,
633
LINEAR EQUATIONS
for 1 F i1 - i 2 - ⭈⭈⭈ - i n F m and 1 F l 1 - l 2 - ⭈⭈⭈ - l n F m. We have n
k
⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b . s⌬ i1 , . . . , i nŽ A . . Ý Ž y1 . .bl k .⌬2,l 1 ,. .. .. ., ,nl k , . . . , l n Ž A . , ks1
Ž 1. where
⌬2,l 1 ,. .. .. ., ,nl k , . . . , l n Ž A . s
al1 2 .. . a l ky 1 2
al1 3 .. . a l ky 1 3
⭈⭈⭈
a l kq 1 2 .. . al n 2
a l kq 1 3 .. . al n 3
⭈⭈⭈
⭈⭈⭈
⭈⭈⭈
al1 n .. . a l ky 1 n a l kq 1 n .. . al n n
.
Since Unq 1Ž A N b . s Ž0. it follows that bl k
al k 1
al k 2
⭈⭈⭈
al k n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
bi 2 .. . bi n
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n s 0, .. . ai n n
or equivalently bl k
al k 1
al k 2
⭈⭈⭈
al k n
0
al k 1
al k 2
⭈⭈⭈
al k n
0
a i11
ai1 2
⭈⭈⭈
ai1 n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
0 .. . 0
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n q bi 2 .. .. . . ai n n bi n
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
a i 2 n s 0. .. . ai n n
Hence 0
al k 1
al k 2
⭈⭈⭈
al k n
bi1
a i11
ai1 2
⭈⭈⭈
ai1 n
bl k⌬ i1 , . . . , i nŽ A . s y bi 2 .. .
ai 2 1 .. . a i n1
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
ai 2 n . .. . ai n n
bi n
634
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Substituting in Ž1. we obtain the chain of equalities ⌬ i1 , . . . , i nŽ A . .⌬1l 1 , . . . , l nŽ A N b .
n
sy
Ý Žy1. k ⌬2l , ,. .. .. ., ,nl , 1
k . . . , ln
ks1
0
a l k1
⭈⭈⭈
al k n
bi1
a i11 .. . a i n1
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . ai n n
Ž A.
⭈⭈⭈
Ž A. . .
. bi n
n
0
n
Ý Žy1. k al 1 ⌬2l , ,. .. .. ., ,nl , k
k . . . , ln
1
ks1
⭈⭈⭈ .. . ⭈⭈⭈
a i11 .. . a i n1
bi n ⌬ l 1 ⭈ ⭈ ⭈ l nŽ A .
0
⭈⭈⭈
0
a i11 .. . a i n1
ai1 2 .. . ai n2
⭈⭈⭈ .. . ⭈⭈⭈
ai1 n .. . ai n n
bi1 sy . .. bi n
k
k . . . , ln
1
Ž A.
ks1
s y bi1 .. .
0
Ý Žy1. k al n ⌬2l , ,. .. .. ., ,nl , ai1 n .. . ai n n
s ⌬ l 1 , . . . , l nŽ A . .⌬1i1 , . . . , i nŽ A N b . .
Note that for t ) 1 we have
n
Ý Ž y1. k al t ⌬ l ,2,. ....,.l, n, k
1
ks1
k
. . . , ln
al1 t
al1 2
⭈⭈⭈
al1 t
⭈⭈⭈
al1 n
al
al 2 2 .. . al n 2
⭈⭈⭈
al 2 t .. . al n t
⭈⭈⭈
al 2 n .. s 0. . al n n
Ž A . s .2 .
t
. al n t
⭈⭈⭈
⭈⭈⭈
Žii. It is clear. PROPOSITION 1.4. Let Ž S . : Ax s b be an o¨ erdetermined system where A is a Ž m = n.-matrix. Suppose that UnŽ A. is a faithful ideal. Then the following statements are equi¨ alent: Ži. Ž S . has a solution Ž necessarily unique. in R. Žii. For j s 1, 2, . . . , n the proportionality
Ž Pj . : ⌬ Ž A . . x j s ⌬ j Ž A N b . has a solution Ž necessarily unique. in R.
635
LINEAR EQUATIONS
In fact ␣ s Ž ␣ 1 , . . . , ␣ n . t is the unique solution of Ž S . if and only if ␣ j is the unique solution of Ž Pj . for j s 1, . . . , n. Proof. Assume Ži.. Suppose that ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ n . t is the solution of Ž S .. For each sequence i1 , i 2 , . . . , i n4 with 1 F i1 - i 2 - ⭈⭈⭈ - i n F m we have that ␣ is solution of the system Ž Si1 , . . . , i n .. So we have AUi1 , . . . , i n . A i1 , . . . , i n . ␣ s AUi1 , . . . , i n .bi1 , . . . , i n hence ⌬ i1 , . . . , i nŽ A . . ␣ j s ⌬ ij1 , . . . , i nŽ A N b . , for j s 1, 2, . . . , n. Consequently ␣ j is the solution of the system Ž Pj .. Assume Žii.. Let ␣ j be the solution of Ž Pj . for j s 1, 2, . . . , n. Consider the element ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ n . t. Since UnŽ A. is a faithful ideal of R then the element
s
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
Ý
ny 1
in
1Fi 1-i 2- ⭈⭈⭈ -i n Fm
⌬ i1 , . . . , i nŽ A .
is a nonzero divisor in the polynomial ring Rw Y x. For k s 1, 2, . . . , m we have in Rw Y x the following chain of equalities:
Ž a k1 ␣ 1 q ⭈⭈⭈ qak n ␣ n . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= ⌬ i1 , . . . , i nŽ A . . Ž a k1 ␣ 1 q ⭈⭈⭈ qa k n ␣ n . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ␣ 1 ⌬ i1 , . . . , i nŽ A . q ⭈⭈⭈ qa k n ␣ n ⌬ i1 , . . . , i nŽ A . . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . . .
636
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Now, since rankŽ A < b . s n it follows that a k1 a i11
bk bi1
ak 2 ai1 2
⭈⭈⭈ ⭈⭈⭈
ak n ai1 n
0 s ai 2 1 .. .
bi 2 .. . bi n
ai 2 2 .. . ai n 2
⭈⭈⭈ .. . ⭈⭈⭈
ai 2 n .. . ai n n
a i n1
s a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . y bk ⌬ i1 , . . . , i nŽ A . . Therefore
Ž a k1 ␣ 1 q ⭈⭈⭈ qak n ␣ n . s
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
Ý
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
= Ž a k1 ⌬1i1 , . . . , i nŽ A N b . q ⭈⭈⭈ qa k n ⌬ni1 , . . . , i nŽ A N b . . s
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
ny 1
in
1Fi 1- ⭈⭈⭈ -i n Fm
s bk
Ý
Y i1qŽ mq1.i 2q ⭈ ⭈ ⭈ qŽ mq1.
bk ⌬ i1 ⭈ ⭈ ⭈ i nŽ A .
ny 1
1Fi 1- ⭈⭈⭈ -i n Fm
in
⌬ i1 ⭈ ⭈ ⭈ i nŽ A . s bk .
Since is a nonzero divisor it follows that a k1 ␣ 1 q ⭈⭈⭈ qa k n ␣ n s bk , and consequently ␣ is a solution of Ž S ..
2. CHARACTERIZATION THEOREMS THEOREM 2.1. Let R be a commutati¨ e ring with unit. Then the following statements are equi¨ alent: Ži. R is an integrally closed ring. Žii. A proportionality Ž P . : a. x s b such that U1Ž a. contains a nonzero di¨ isor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. An o¨ erdetermined system Ž S . : Ax s b, where A is an Ž m = n.matrix such that UnŽ A. contains a nonzero di¨ isor, has a solution in R if and only if Up Ž A. s Up Ž A < b . for p s n, n q 1.
637
LINEAR EQUATIONS
Proof. First we prove that Ži. and Žii. are equivalent. Assume Ži.. Let Ž P . : a. x s b be a proportionality such that U1Ž a. contains a nonzero divisor and suppose that rankŽ a. s rankŽ a < b . and U1Ž a. s U1Ž a < b .; we show that Ž P . has a solution in R. By Lemma 1.2, the proportionality Ž P . has a solution rs in the total quotient ring T Ž R ., because rankŽ a. s rankŽ a < b . s 1. Condition U1Ž a. s U1Ž a < b . assures that there exist elements c i j g R, 1 F i, j F n, such that bi s Ý njs1 c i j a j . Consequently in T Ž R . we have the equalities: a1 . a2 .
s
s
s c11 a1 q c12 a2 q ⭈⭈⭈ qc1 n a n s c 21 a1 q c 22 a2 q ⭈⭈⭈ qc2 n a n .. .
an .
s
s c n1 a1 q c n2 a2 q ⭈⭈⭈ qc n n a n .
If C is the Ž n = n.-matrix C s Ž c i j ., then the above equalities can be rewritten in the form
ž
s
.Id n y C a s 0,
/
where Id n is the identity matrix of order n. Therefore det
ž
s
.Id n y C .a i s 0,
/
for i s 1, 2, . . . , . Since U1Ž a. contains a nonzero divisor it follows that 0 s det
ž
s
.Id n y C s
n
/ ž / s
q c ny1
ny1
q ⭈⭈⭈ qc1
ž / s
ž / s
q c0 ,
where c i g R for i s 0, 1, . . . , n y 1. Thus s is integer over R and hence g R because R is an integrally closed ring. s Assume Žii.. Let rs be an element of T Ž R . such that
ž / s
n
q c ny 1
ž / s
ny1
q ⭈⭈⭈ qc1
ž / s
q c 0 s 0,
where c i g R for i s 0, 1, . . . , n y 1. Consider the system of linear equa-
638
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
tions Ž P . : ax s b given by
¡s
Ž P . :~
x x
s s
ny 2 sx ny 1 x
s s
ny 1
s
ny 2
.. .
¢
s ny2 2 s ny3 .. . y Ž c0 s
ny1
ny1 q c1 s ny2 q ⭈⭈⭈ qc ny1 ny1 . .
For Ž P . one has rank Ž a . s rank Ž a < b . s 1, U1 Ž a . s U1 Ž a < b . s Ž s ny 1 , s ny2 , . . . , ny2 s, ny1 . . Moreover U1Ž a. contains the nonzero divisor s ny 1. But Žii. the proportionality Ž P . has a unique solution in R that obviously is also the unique solution in T Ž R .. Since is the solution of Ž P . in T Ž R . it follows that s
g R. Next we prove that Žii. and Žiii. are equivalent. Clearly Žii. implies Žiii.. Conversely, let A be an Ž m = n.-matrix such that UnŽ A. contains a nonzero divisor. Suppose that the overdetermined system Ž S . : Ax s b satisfies Up Ž A. s Up Ž A N b . for p s n, n q 1 Ži.e., rankŽ A. s rankŽ A < b . s n and UnŽ A. s UnŽ A N b .. Consider the proportionalities s
Ž Pj . : ⌬ Ž A . x j s ⌬ j Ž A N b . ,
j s 1, 2, . . . , n
introduced in Section 1. By Lemma 1.3, we have rank Ž ⌬ Ž A . . s rank Ž ⌬ Ž A . < ⌬ j Ž A N b . . s 1, and U 1 Ž ⌬ Ž A . . s U1 Ž ⌬ Ž A . < ⌬ j Ž A N b . .
for j s 1, 2, . . . , n.
Moreover U1Ž ⌬Ž A.. s UnŽ A. contains a nonzero divisor. By Žiii. the proportionality Ž Pj . for j s 1, 2, . . . , n has a solution in R and hence, by Proposition 1.4, the system Ž S . has a solution in R. COROLLARY 2.2. are equi¨ alent:
Let R be an integral domain. The following statements
Ži. R is an integrally closed domain. Žii. A proportionality Ž P . : ax s b has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. An o¨ erdetermined system Ž S . : Ax s b, where A is an Ž m = n.matrix has a solution in R if and only if Up Ž A. s Up Ž A N b . for p s n, n q 1.
639
LINEAR EQUATIONS
THEOREM 2.3. equi¨ alent:
Let R be a commutati¨ e ring. The following statements are
Ži. The local ring R ᒊ is an integrally closed domain for e¨ ery maximal ᒊ of R. Žii. A proportionality Ž P . : ax s b has a solution in R if and only if Ž rank a. s rankŽ a < b . and Up Ž a. s Up Ž a < b . for p s 1, 2. Proof. Assume Ži.. Let Ž P . : ax s b be a proportionality such that Up Ž a. s Up Ž a < b . for p s 1, 2. By Žw5, Proposition 1x. the proportionality Ž P . has a solution in R if and only if Ž Pᒊ . has a solution in R ᒊ for every maximal ideal ᒊ of R. Let ᒊ be a maximal ideal of R. If U1Ž a.. R ᒊ s U1Ž a < b .. R ᒊ s 0 then Ž Pᒊ . is the trivial system of linear equations Ž Pᒊ . : 0. x s 0 and hence Ž Pᒊ . has a solution in R ᒊ . On the other hand if U1Ž a.. R ᒊ s U1Ž a < b .. R ᒊ / 0 then the proportionality
Ž Pᒊ . :
a 1
.x s
b 1
satisfies a
Up
ž / 1
s Up Ž a . . R ᒊ s Up Ž a < b . . R ᒊ s Up
a b
ž / 1 1
for p s 1, 2.
By Corollary 2.2 Ž Pᒊ . has a solution in R ᒊ . Assume Žii.. Let ᒊ be a maximal ideal of R. First we prove that R ᒊ is a u ¨ domain. Suppose that there are two elements u, ¨ in R such that 1 , 1 are u
¨
nonzero elements in R ᒊ and 1 . 1 s 0 or equivalently s.u.¨ s 0 for some s f ᒊ. Then the proportionality Ž P . : a. x s b given by xs0 Ž P . : s.u. ¨ . x s s.¨
½
satisfies rankŽ a. s rankŽ a < b . s 1 and U1Ž a. s U1Ž a < b . s Ž u, s.¨ .. However Ž P . has no solution in R because Ž Pᒊ . has no solution in R ᒊ , see Žw3, Theorem 6x.. Next we prove that R ᒊ is an integrally closed domain. Let Ž P˜. : ax ˜ s ˜b by a proportionality in R ᒊ given by
Ž P˜. :
½
ai si
.x s
bi ti
5
, 1FiFm
such that Up Ž a ˜. s Up Ž a˜< ˜b . for p s 1, 2. We show that Ž P˜. has a solution in R ᒊ .
640
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Since U2 Ž a ˜< ˜b . s Ž0. there exists s f ᒊ such that s Ž t i .a i .s j .bj y si .bi .t j .a j . s 0 for 1 F i , j F m. Moreover there exists t f ᒊ such that t.si .bi g Ž t 1 .a1 , t 2 .a2 , . . . , t m .a m . , because U1Ž a ˜. s U1Ž a˜< ˜b .. It follows that the proportionality Ž P . : ax s b over R given by
Ž P . : t i .ai . x s s.t.si .bi 4 1FiFm satisfies Up Ž a. s Up Ž a < b . for p s 1, 2. By Ži. Ž P . has a solution ␣ in R and ␣ hence s.t is the solution of Ž P˜.. 3. WHEN IS THE POLYNOMIAL RING Rw T x INTEGRALLY CLOSED? PROPOSITION 3.1. grally closed. Then Ži. Žii. ideal has Žiii.
Let R be a commutati¨ e ring such that Rw T x is inte-
R is a reduced ring. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. R is integrally closed.
Proof. Ži. If a is a nonzero nilpotent element of R then
a T
is an
integral over R and however T f Rw T x. Žii. Since U1Ž a. is a faithful ideal of R it follows that U1Ž a. Rw T x contains a nonzero divisor in Rw T x. By Theorem 2.1, Ž P . has a solution in Rw T x. Therefore Ž P . has a solution in R because the extension R ª Rw T x is faithfully flat. Žiii. This is a straightforward application of Theorem 2.1 and the above statement. a
The main result of this section establishes that Ži. and Žii. are sufficient conditions to assure that Rw T x is integrally closed. Before starting the main result of this section we need some previous results:
641
LINEAR EQUATIONS
LEMMA 3.2. Let R be a reduced ring and Ž P . : f. x s g be a proportionality in Rw T x gi¨ en by
¡ f ŽT . . x s g ŽT . 1
Ž P . :~
¢f
1
f2 Ž T . . x s g2 Ž T . .. . m
Ž T . . x s gm Ž T . .
Suppose that U1Ž f . is a faithful ideal. If ␣ ŽT . is a solution of Ž P . then
⭸ ⬚␣ Ž T . F max ⭸ ⬚g 1 Ž T . , ⭸ ⬚g 2 Ž T . , . . . , ⭸ ⬚g m Ž T . 4 , where ⭸ ⬚␣ ŽT . denotes the degree of the polynomial ␣ ŽT .. Proof. Put f i Ž T . s a i0 q a i1T q ⭈⭈⭈ qa i p i T p i , g i Ž T . s bi0 q bi1T q ⭈⭈⭈ qbi q i T q i , for i s 1, 2, . . . , m. Let ␣ ŽT . s ␣ 0 q ␣ 1T q ⭈⭈⭈ q␣ r T r be the solution of Ž P .. Suppose that r ) qi for i s 1, 2, . . . , m. Then we have the equalities ai p i ␣ r s 0 a i p iy 1 ␣ r q a i p i ␣ ry1 s 0 .. . a i0 ␣ r q a i1 ␣ ry1 q ⭈⭈⭈ s 0, for i s 1, 2, . . . , m. It follows that a i p i ␣ rsa i p iy1 ␣ r2s ⭈⭈⭈ sa i0 ␣ rrq1 s 0,
1FiFm.
f i ŽT .. ␣ rrq1
Hence s 0 for i s 1, . . . , m and consequently U1Ž f .. ␣ rrq1 s 0. Since R is reduced we have that ␣ rrq1 / 0. This provides the contradiction that we were seeking because, by hypothesis, U Ž f . is faithful ideal. Let f ŽT . s a0 q a1T q ⭈⭈⭈ qa p T p be an element of Rw T x. We denote ᑾ f the content of f ŽT ., Ži.e., the ideal of R generated by the coefficients of f ŽT ... For each positive integer s we denote by AŽ f ; s . the ŽŽ p q s . = s .matrix
¡a
¦
a1 a2 A Ž f ; s . s .. .
0 a0 a1 .. .
0 0 a0 .. .
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
0 0 0 .. .
0 0 0 .. .
0
0
0
⭈⭈⭈
ap
a py1
0
0
⭈⭈⭈
0
ap
0
¢0
.
§
642
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
LEMMA 3.3.
For a positi¨ e integer r with 1 F r F s we ha¨ e Ur Ž A Ž f ; s . . s ᑾ rf .
Proof. We use induction on r. When r s 1 we have the equality U1 Ž A Ž f ; s . . s ᑾ f , for each positive integer s. We now suppose that r ) 1 and that Ury1 Ž A Ž f ; s . . s ᑾ ry1 , f for each positive integer s such that r y 1 F s. Clearly Ur Ž AŽ f ; s .. is contained in ᑾ rf . By the induction hypothesis we have Ury1 Ž A Ž f ; s . . s ᑾ f .U Ury1 Ž A Ž f ; s y 1 . . . s ᑾ f .U ᑾ rf s ᑾ f .ᑾ ry1 f Next we prove, by induction of k, that a k Ury1 Ž A Ž f ; s y 1 . . : Ur Ž A Ž f ; s . . , for k s 0, 1, . . . , n. Consider the matrix AŽ f, s y 1. as a submatrix of AŽ f, s . by
¡a
0
0
⭈⭈⭈
¦
0
a1 .. .
AŽ f ; s . s a p 0 .. .
¢0
.
AŽ f ; s y 1.
§
So it is clear that a0 Ury1 Ž A Ž f ; s y 1 . . : Ur Ž A Ž f ; s . . . Let k ) 0 and suppose that a s Ury1 Ž A Ž f ; s y 1 . . s a s Ury1 Ž A Ž f ; s . . : Ur Ž A Ž f ; s . . , ry 1 be the submatrix of for all s F k y 1. Let AŽ f, s y 1. ij11,, .. .. .. ,, ijry AŽ f, s . 1 composed by the rows i1 , i 2 , . . . , i ry1 with 2 F i1 - i 2 - ⭈⭈⭈ - i ry1 F p q s and the columns j1 , j2 , . . . , jry1 with 2 F j1 - j2 - ⭈⭈⭈ - jry1 F s. We con-
643
LINEAR EQUATIONS
sider the matrix
¡a
a kyj1q1
k
¦
⭈⭈⭈
a kyj 2q1
a i1q1 B s a i q1 2 .. . .. .
j ,..., j
ry 1 A Ž f ; s y 1 . i11, . . . , i ry 1
¢
,
§
where for a positive integer i we put ayi s a pqi s 0. Note that if k q 1 is j1 , . . . , j ry 1 different of i1 , i 2 , . . . , i ry1 then B is the submatrix of AŽ f ; s .1,kq1, i 1 , . . . , i ry 1 . If k q 1 is equal to i s for some s then the determinant of B is zero and hence detŽ B . s 0 g Ur Ž AŽ f ; s ... Therefore ry1
j , . . . , ju
Ý akyj q1 . ␦t g Ur Ž AŽ f ; s . . ,
det Ž B . s a k .det A Ž f ; s y 1 . i11, . . . , i ryry1 1 q
ž
/
t
ts1
where ␦ t is the adjoint on B associated to the element a kyj tq1 of the first row. Consequently ␦ t g Ury1Ž AŽ f ; s .. and hence j ,..., j
ry 1 det Ž B . s a k det A Ž f ; s y 1 . i11, . . . , i ry 1
ž
/
Ury1 Ž A Ž f ; s . . . q terms in Ž a0 , a1 , . . . , a ky1 . .U Since detŽ B . g Ur Ž AŽ f ; s .. and, by the induction hypothesis,
Ž a0 , . . . , ary1 . Ury1 Ž A Ž f ; s . . : Ur Ž A Ž f ; s . . , it follows that j ,..., j
ry 1 a k .det A Ž f ; s y 1 . i11, . . . , i ry g Ur Ž A Ž f ; s . . . 1
ž
/
COROLLARY 3.4. Let f 1ŽT ., f 2 ŽT ., . . . , f mŽT . be elements of Rw T x and let s be a positi¨ e integer. If AŽ f 1 , f 2 , . . . , f m ; s . is the block matrix AŽ f 1 ; s . AŽ f 1 , f 2 , . . . , fm ; s . s
0
AŽ f 2 ; s . , .. .
AŽ fm ; s .
644
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
then Ur Ž A Ž f 1 , f 2 , . . . , f m ; s . . s
Ý
r1qr 2q ⭈⭈⭈ qr m sr
ᑾ rf 11 ᑾ rf 22 ⭈⭈⭈ ᑾ rf mm ,
for 1 F r F s. Proof. Let pi be the degree of f i for i s 1, 2, . . . , m. For a positive integer s we have that AŽ f 1 , f 2 , . . . , fm ; s . s AŽ f ; s . , where f ŽT . is the polynomial f Ž T . s f 1 q T p 1qs f 2 q T p 1qp 2q2 s f 3 q ⭈⭈⭈ qT p 1qp 2q ⭈⭈⭈ qp my 1qŽ my1. s f m , since
ᑾ f s ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m ,
the result follows from the above lemma. We denote by RŽT . the quotient ring Sy1 Rw T x of the polynomial ring Rw T x where S is the multiplicative closed set of f g Rw T x such that ᑾ f s R. LEMMA 3.5. Let R be a ring and let Y be an element of RŽT . that is integral o¨ er Rw T x. Then there exists h g Rw T x such that Y y h is nilpotent in RŽT .. Proof. See Žw6, Lemma 16.2x.. The main result of this section is: THEOREM 3.6. Let R be a commutati¨ e ring. Then the following statements are equi¨ alent: Ži. Rw T x is an integrally closed ring. Žii. R satisfies the following properties: ᎏ R is a reduced ring. ᎏ A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. R satisfies the following properties: ᎏ R is a reduced ring. ᎏ An o¨ erdetermined system Ž S . : A. x s b where A is an Ž m = n.-matrix such that UnŽ A. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s n, n q 1. Proof. Ži. implies Žii. by Proposition 3.1. The equivalence between Žii. and Žiii. follows from Proposition 1.3. We show that Žiii. implies Ži. using Theorem 2.1
645
LINEAR EQUATIONS
Suppose that
¡f ŽT . . xŽT . s g ŽT . 1
Ž P . :~
1
f2 Ž T . . x Ž T . s g2 Ž T . .. .
¢f
m
Ž T . . x Ž T . s gm Ž T .
is a proportionality, denoted by Ž P . : f. x s g, in Rw T x such that U1Ž f . contains a nonzero divisor in Rw T x and Up Ž f . s Up Ž f < g . for p s 1, 2. Let m
FŽT . s
Ý i Ž T . . f i Ž T . g U1 Ž f Ž T . . is1
be a nonzero divisor, and consider m
GŽ T . s
Ý i Ž T . . g i Ž T . . is1
GŽT . is the solution of Ž P . in T Ž Rw T x.. Moreover s F Ž T . is an integer over Rw T x, see Theorem 2.1 Put f i Ž T . s a i0 q a i1T q ⭈⭈⭈ qa i p i T p i ,
Then s
GŽ T . F ŽT .
g i Ž T . s bi0 q bi1T q ⭈⭈⭈ qbi q i T q i , for i s 1, 2, . . . , m and q s max q1 , q2 , . . . , qm 4 . By Lemma 3.2, if Ž P . has a solution x ŽT . Žnecessarily unique. in Rw T x then x ŽT . is the form x ŽT . s x 0 q x 1T q ⭈⭈⭈ qx q T q. Consequently the proportionality Ž P . has a solution in Rw T x if and only if the system of linear equations over R,
Ž S . : AŽ f 1 , f 2 , . . . , fm ; q q 1. x s b has a solution in R, where AŽ f 1 , f 2 , . . . , f m ; q q 1. is defined in Corollary 3.4, x s Ž x 0 , x 1 , . . . , x q . t is the column vector of indeterminates and b is the column vector b s Ž b10 , . . . , b1 q1 , 0, . . . , 0, b 20 , . . . , b 2 q 2 , t
0, . . . , 0, . . . , bm 0 , . . . , bm q m , 0, . . . , 0 . , Next we show that Ž S . satisfies the conditions of statement Žiii.. By Corollary 3.4, we have Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s
Ý
r1qr 2q ⭈⭈⭈ qr m sqq1
ᑾ rf 11 ᑾ rf 22 ⭈⭈⭈ ᑾ rf mm
s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
qq 1
.
646
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
Since the ideal of Rw T x generated by f 1ŽT ., f 2 ŽT ., . . . , f mŽT . contains a nonzero-divisor it follows that ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m is a faithful ideal of R. The hypothesis U1Ž f . s U1Ž f < g . implies that ᑾ g i : ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m , for i s 1, 2, . . . , m. So we have Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . : Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . : Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . q Ž ᑾ g 1 q ᑾ g 2 q ⭈⭈⭈ qᑾ g m . Uq Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
qq 1
q Ž ᑾ g 1 q ᑾ g 2 q ⭈⭈⭈ qᑾ g m . . Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m . s Ž ᑾ f 1 q ᑾ f 2 q ⭈⭈⭈ qᑾ f m .
q
qq 1
s Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . . Therefore Uqq 1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . . s Uqq1 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . . Finally we prove that Uqq 2 Ž AŽ f 1 , f 2 , . . . , f m ; q q 1.< b . s Ž0.. Consider the canonical inclusion of R in R⬘ s T Ž Rw Z x. where Z is a new indeterminate and note that the proportionality Ž P . has a solution in R⬘w T x if and only if the system Ž S . has a solution in R⬘. Next we show that Ž P . has a solution in R⬘w T x. Replacing T by T q Z, we obtain the proportionality
¡ f ŽT . . xŽT . s g ŽT .
Ž P⬘ . :~
¢f
X 1 f 2X
X m
X 1 g X2
ŽT . . xŽT . s ŽT . .. .
Ž T . . x Ž T . s g Xm Ž T . ,
where f iX ŽT . s f i ŽT q Z . and g Xi ŽT . s g i ŽT q Z . for i s 1, 2, . . . , m. The unique solution of Ž P⬘. in T Ž R⬘w T x. is ⬘ s G⬘ŽT .rF⬘ŽT . where F⬘ŽT . s F ŽT q Z . and G⬘ŽT . s GŽT q Z .. Moreover ⬘ is integral over R⬘w T x. Since F Ž Z . is a nonzero divisor on Rw Z x it follows that F⬘Ž0. s F Ž Z . is a
LINEAR EQUATIONS
647
unit in R⬘ and hence ⬘ s G⬘ŽT .rF⬘ŽT . g R⬘ŽT .. By Lemma 3.5 there exists h g R⬘w T x such that ⬘ y h is nilpotent in R⬘ŽT .. Consequently ⬘ s h g R⬘w T x because R is a reduced ring. Thus Ž P⬘. has a solution in R⬘w T x and therefore Ž P . has a solution in R⬘w T x. Since Ž P . has a solution in R⬘w T x it follows that Ž S . has a solution in R⬘. Consequently Uqq 2 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . . R⬘ s Ž 0 . , and hence Uqq 2 Ž A Ž f 1 , f 2 , . . . , f m ; q q 1 . < b . s Ž 0 . , because R is contained in R⬘. The system Ž S . satisfies the conditions of statement Žiii.. Thus Ž S . has a solution in R and hence has solution in Rw T x. Recall that the set Spec R of all prime ideals of R together with its Zariski topology is quasi-compact. However, the subset Min R of all minimal prime ideals of R in general is not quasi-compact. LEMMA 3.7. Let R be a reduced ring such that Min R is quasi-compact. If ᑾ is a finitely generated ideal of R then there exists a finitely generated ideal ᑿ : AnnŽ ᑾ . such that ᑾ q ᑿ is a faithful ideal. Proof. Let ᑾ be a finitely generated ideal of R. Since R is a reduced then Rw T x satisfies the annihilator condition, see Žw6, pp. 5 and 7x.. Hence there exists a polynomial g ŽT . g Rw T x such that Ann Ž ᑾ . R w T x . s Ann Ž g Ž T . . . The canonical homomorphism i: R ª Rw T x induces a homeomorphism between Min R and Min Rw T x. Consequently Min Rw T x is quasi-compact. By Žw6, Theorem 4.3x. there exist polynomials h1ŽT ., h 2 ŽT ., . . . , h k ŽT . in AnnŽ g ŽT .. such that Ann Ž h1 Ž T . , h 2 Ž T . , . . . , h k Ž T . , g Ž T . . s 0. Let ᑿ be the ideal of R generated by all coefficients of the polynomials h1ŽT ., h 2 ŽT ., . . . , h k ŽT .. Then we have that ᑿ : AnnŽ ᑾ ., because h i ŽT . : Ann Ž g Ž T .. s Ann Ž ᑾ R Ž T .. , and Ann Ž ᑾ q ᑿ . s Ž 0 . , because AnnŽ h1ŽT ., h 2 ŽT ., . . . , h k ŽT .. l AnnŽ ᑾ RŽT .. s Ž0.. COROLLARY 3.8. Let R be a reduced ring such that with Min R quasicompact. Then the following statements are equi¨ alent: Ži. A proportionality Ž P . : a. x s b o¨ er R has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2.
648 Žii. Žiii. Živ. ideal has
DEFRANCISCO-IRIBARREN AND HERMIDA-ALONSO
R ᒊ is an integrally closed domain for e¨ ery ᒊ maximal of R. Rw T x is an integrally closed ring. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2.
Proof. By Theorems 2.3 and 3.1 it is sufficient to prove that Živ. implies Ži.. Let Ž P . : a. x s b be a proportionality in R such that Up Ž a. s Up Ž a < b . for p s 1, 2. By Lemma 3.7 there exist c i g AnnŽ Ui Ž a.., i s 1, 2, . . . , t such that Ann Ž ² c1 , c 2 , . . . , c k , a1 , a2 , . . . , a n : . s 0. The proportionality
¡a . x s b 1
Ž P⬘ . :~
1
a2 . x s b 2 .. . a n . x s bn , c1 . x s 0 c2 . x s 0 .. .
¢c . x s 0 k
by Živ. has a solution in R and hence Ž P . has a solution in R. Recall that R satisfies Property A if each finitely generated ideal of R composed of zero divisors has nonzero annihilator. Equivalently R satisfies Property A if and only if each generated faithful ideal of R contains a nonzero divisor. COROLLARY 3.9. Let R be a reduced ring satisfying Property A. Then the following statements are equi¨ alent: Ži. Rw T x is an integrally closed ring. Žii. A proportionality Ž P . : a. x s b o¨ er R such that U1Ž a. is a faithful ideal has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Žiii. A proportionality Ž P . : a. x s b o¨ er R such that Ui Ž a. contains a nonzero di¨ isor has a solution in R if and only if Up Ž a. s Up Ž a < b . for p s 1, 2. Živ. R is an integrally closed ring.
649
LINEAR EQUATIONS
REFERENCES 1. T. Akiba, Integrally-closedness of polynomial rings, J. Math. Japan 6 Ž1980., 67᎐75. 2. P. Camion, L. S. Levi, and H. B. Mann, Linear equations over a commutative ring, J. Algebra 19 Ž1972., 432᎐446. 3. J. A. Hermida-Alonso and T. Sanchez-Giralda, Linear equations over commutative rings ´ and determinantal ideals, J. Algebra 99, No. 1 Ž1986., 72᎐79. 4. J. A. Hermida-Alonso, Linear equations over commutative rings and grade theory, J. Algebra 148, No. 2 Ž1992., 497᎐503. 5. J. A. Hermida-Alonso, Linear equations over commutative rings and finite free resolutions, J. Algebra 164, No. 2 Ž1994., 452᎐467. 6. J. A. Huckaba, ‘‘Commutative Rings with Zero Divisors,’’ Dekker, New York, 1988. 7. T. Lucas, Characterizing when Rw X x is integrally closed, Proc. Amer. Math. Soc. 105 Ž1989., 861᎐867. 8. T. Lucas, Characterizing when Rw X x is integrally closed II, J. Pure Appl. Algebra 61 Ž1989., 49᎐52. 9. T. Lucas, The complete integral closure of Rw X x, Trans. Amer. Math. Soc. 330 Ž1992., 757᎐768. 1 10. N. H. McCoy, Rings and ideals, Carus Math. Monographs 8, No. Ž1948.. 4
11. D. G. Northcott, ‘‘Finite Free Resolutions,’’ Cambridge Univ. Press, Cambridge, U.K., 1976. 12. Y. Quentel, Sur la compacite ´ du spectre minimal d’un anneau, Bull. Soc. Math. France 99, No. 18 Ž1971., 265᎐272. 13. D. W. Sharpe, Grade and the theory of linear equations, Linear Algebra Appl. 18 Ž1977., 25᎐32. 14. W. V. Vasconcelos, Annihilators of modules with a finite free resolution, Proc. Amer. Math. Soc. 29 Ž1971., 440᎐442.