CODIMENSION AND REGULARITY OVER COHERENT SEMILOCAL RINGS

Transcription

1 COMMUNICATIONS IN ALGEBRA, 29(10), (2001) CODIMENSION AND REGULARITY OVER COHERENT SEMILOCAL RINGS Gaoha Tang, 1,2 Zhaoyong Hang, 1 and Wenting Tong 1 1 Department of Mathematics, Nanjing University, Nanjing , P. R. China 2 Department of Mathematics, Gangxi Teachers College, Nanning , P. R. China ABSTRACT In this paper, the reslts on codimension and reglarity over noetherian local rings and coherent local rings are extended to coherent semilocal rings and some sefl examples of coherent semilocal rings are constrcted Mathematics Sbject Classification: 13H99, 16E10. INTRODUCTION Throghot this paper it is assmed that all rings are commtative coherent rings with identity and all modles are nitary. Or aim in this paper is to extend the reslts on codimension and reglarity which were stdied in [7] and [8] for noetherian local rings and coherent local rings to coherent semilocal rings Copyright # 2001 by Marcel Dekker, Inc.

2 4812 TANG, HUANG, AND TONG Let R be a ring and M an R-modle. Recall that M is called finitely presented if there is a finitely generated R-modle P and a finitely generated sbmodle N of P sch that P=N M. R is called a coherent ring if every finitely generated ideal of R is finitely presented. R is called a reglar ring if every finitely generated ideal of R has finite projective dimension ([5]). In case R is a noetherian ring, the notion of reglarity given here coincides with that in [7]. In x 2, we stdy the codimension of modles, and show that if R is a reglar coherent semilocal ring and every maximal ideal of R is finitely generated, then m-codim R ðrþ ¼Codim Rm ðr m Þ¼w:gl:dim R m for every maximal ideal m of R, andcodim R ðrþ ¼w:gl:dim R. x 3 deals with the reglarity of coherent semilocal rings. In [7] Theorem 60 and Theorem 69, Kaplansky proved that a noetherian local ring R is reglar if and only if the niqe maximal ideal of R is generated by a reglar R-seqence. In [8], we proved that a coherent local ring R whose maximal ideal m is finitely generated is reglar if and only if m is generated by a reglar R-seqence (see [8] Theorem 2.6). The main reslt of this section is that if R is an indecomposable coherent semilocal ring whose every maximal ideal is finitely generated, then R is reglar if and only if every maximal ideal of R is generated by a reglar R-seqence if and only if every maximal ideal m which satisfies w:gl:dim R ¼ w:gl:dim R m is generated by a reglar R-seqence if and only if there exists a maximal ideal m of R sch that w:gl:dim R ¼ w:gl:dim R m and m is generated by a reglar R-seqence. In x 4 we provide an example of a non-noetherian indecomposable coherent semilocal ring R with exactly t maximal ideals and with weak global dimension eqal to s, for any natral nmbers t and s. In this paper, we se J; MaxðRÞ; gl:dim R; w:gl:dim R; pd R ðmþ; fd R ðmþ; id R ðmþ; Codim R ðmþ; FP-id R ðmþ; R M; f:g: R M; f:r: R M for the Jacobson radical, the maximal spectrm, global dimension, weak global dimension of R, projective dimension, flat dimension, injective dimension, codimension, FP-injective dimension of R-modle M, the category of R- modles, the category of finitely generated R-modles and the category of finitely presented R-modles, respectively. 1 PRELIMINARIES Definition 1. Let R be a ring and I an ideal of R, M 2 R M. A finite seqence a 1 ;...; a n 2 I is called a reglar M-seqence if a i is not a zero divisor for the modle M=ða 1 ;...; a i 1 ÞM, i ¼ 1;...; n, and M 6¼ ða 1 ;...; a n ÞM. We define I-codim R ðmþ ¼spft j a 1 ;...; a t is a reglar M-seqence in Ig and

3 CODIMENSION AND REGULARITY 4813 Codim R ðmþ ¼spfI-codim R ðmþ ji is an ideal of Rg which are called the codimension of M in I and codimension of M respectively. Since every ideal of R mst be contained in a maximal ideal of R, we have Codim R ðmþ ¼spfmcodim R ðmþ jm 2 MaxðRÞg. Particlarly Codim R ðrþ ¼spfm-codim R ðrþjm 2 MaxðRÞg. Definition 2. Let R be a ring, M 2 R M. The FP-injective dimension of M, denoted by FP-id R ðmþ, is eqal to the least integer n 0 forwhich ExtR nþ1 ðp; MÞ ¼0 for every finitely presented R-modle P. If no sch n exists set FP-id R ðmþ ¼1. In this section, we shall give some lemmas which will be sed later. Lemma 1.1 ([5] Corollary 2.5.5). Let R be a coherent ring, M 2 f :r: R M. Then pd R ðmþ ¼fd R ðmþ. Lemma 1.2 ([6] Theorem 5). If R is a coherent semilocal ring, then w:gl:dim R ¼ fd R ðr=jþ ¼id R ðr=jþ ¼FP-id R ðr=jþ: Lemma 1.3. Let R be a ring, m 2 MaxðRÞ, M 2 R M.Ifa 1 ;...; a n 2 m is a reglar M-seqence, then the seqence a 1 ;...; a n, considered as elements in R m, is a reglar M m -seqence. Proof. See the proof of Proposition 1.2 in [1]. Lemma 1.4. Let R be a coherent local ring, M 2 f :r: R M. Then Codim R ðmþ w:gl:dim R: Proof. Let m be the maximal ideal of R and let a 1 ;...; a s 2 m be any reglar M-seqence in m. By [8] Lemma 2.3 we have pd R ðm=ða 1 ;...; a s ÞMÞ ¼ pd R ðmþþs. Since M is finitely presented, so is M=ða 1 ;...; a s ÞM. Therefore s pd R ðm=ða 1 ;...; a s ÞMÞ ¼fd R ðm= ða 1 ;...; a s ÞMÞ w:gl:dim R. Ths we have Codim R ðmþ w:gl:dim R. Corollary 1.5. Let R be a coherent ring, M 2 f :r: R M. Then Codim R ðmþ w:gl:dim R: Proof. By Lemma 1.3 and Lemma 1.4, we have Codim R ðmþ ¼spfm-codim R ðmþjm 2 MaxðRÞg spfcodim Rm ðm m Þjm 2 MaxðRÞg spfw:gl:dim R m j m 2 MaxðRÞg ¼ w:gl:dim R. Lemma 1.6. Let R be a semilocal domain bt not a field.

4 4814 TANG, HUANG, AND TONG (1) If m 1 2 MaxðRÞ and m 1 is finitely generated, then there exists an irredcible element a of R in m 1 ; (2) If m 1 2 MaxðRÞ and a 2 m 1 is an irredcible element of R, then R=ðaÞ is a local ring with niqe maximal ideal m 1 ¼ m 1 =ar: (3) If m 2 MaxðRÞ is finitely generated and a 2 m is an irredcible element of R, then a, considered as an element in R m, can extended to a minimal set of generators of the niqe maximal ideal mr m of R m. Proof. (1) We Assme that MaxðRÞ ¼fm 1 ; m 2 ;...; m t g and m i 6¼ m j, 8i 6¼ j. We only need to prove that there exists a 2 m 1 sch that a =2 m i m j, 81 i; j t: Since m 1 ; m 2 ;...; m t are all different maximal ideals of R, m 1 m i ¼ m 1 \ m i 7 Consider the natral homomorphism f : R! R m1, jðrþ ¼ r 1. Since R is a domain, j is injective. If m 2 1 ¼ m 1, then ðm 1 R m1 Þ 2 ¼ m 2 1 R m 1 ¼ m 1 R m1. Since m 1 is finitely generated, so is m 1 R m1. It follows from Nakayama s Lemma that m 1 R m1 ¼ 0, so m 1 ¼ 0 and R is a field, which is a contradiction. Therefore m m 1, 82 i t. -- m 1 and every m1mið1 i tþ is a proper sbmodle of m1 as an R-modle and S t i¼1 m 1m i 7 a 62 m 1 m i, 81 i t. Ifa 2 m i m j, i; j 6¼ 1, then a ¼ a i a j, a i 2 m i, a j 2 m j and a i ; a j 62 m 1.Soa¼a i a j 62 m 1, which is a contradiction. Therefore a 62 m i m j, 81 i; j t. (2) It is easy to see that MaxðR=aRÞ ¼fm=aR j m 2 T MaxðRÞ; a 2 mg. Since a is an irredcible element of R, a 62 m 1 m i ¼ m 1 mi, i ¼ 2;...; t. So a 62 m i, i ¼ 2;...; t, and hence MaxðR=aRÞ ¼fm 1 =arg and R=aR is a local ring. (3) Since a is an irredcible element of R, a is also an irredcible -- m 1. Ths there exists a 2 m1 sch that element of R m and a 62 ðmr m Þ 2. From [3] x 8.3 Exercise 1 we know that a can be extended to a minimal set of generators of mr m. Lemma 1.7 ([3] P. 299, Theorem 2). If R is a semilocal ring, then R has an indecomposable decomposition R ¼ R 1 R n sch that every R i ð1 i nþ is an indecomposable semilocal ring. Let R be a ring and R ¼ R 1 R n an indecomposable decomposition of R. Then every maximal ideal of R has the form m ¼ R 1 R i 1 m i R iþ1 R n ð1þ where m i 2 MaxðR i Þ, and we have a natral ring isomorphism R m R imi. Lemma 1.8. Let R be a coherent local ring with maximal ideal m. Ifm is finitely generated, then R is reglar if and only if w:gl:dim R < 1 if and only if

5 CODIMENSION AND REGULARITY 4815 m is generated by a reglar R-seqence. In addition, if m is generated by a reglar R-seqence with q elements, then Codim R ðrþ ¼w:gl:dim R ¼ q. Proof. See [8] Theorem 2.1, Theorem 2.6 and Corollary 2.7. Lemma 1.9. Let R be a reglar coherent local ring with maximal ideal m.ifmis finitely generated and a; a 2 ;...; a n is a minimal set of generators of m, then w:gl:dim R=aR ¼ w:gl:dim R 1 < 1: In this case R=aR is also a reglar coherent local ring. Proof. By Lemma 1.8, we have w.gl.dim R < 1: Set R ¼ R=aR, m ¼ m=ar, A ¼ ar, B ¼ am þ a 2 R þþa q R.Itis easy to verify that m ¼ A þ B; A \ B ¼ am: So m=am ¼ðAþBÞ=am ða=amþðb=amþ and hence m ¼ m=ar ðm=amþ=ðar=amþ¼ðm=amþ=ða=amþ B=am; that is, m is isomorphic to a direct smmand of m=am as an R-modle, and so as an R-modle. Therefore pd Rð mþ pd Rðm=amÞ: It follows from [11] Corollary 5 that R is a GCD domain, then a is not a zero divisor on m and R, and ths by [5] Theorem we have that pd Rð mþ pd Rðm=amÞ ¼pd R=aR ðm=amþ ¼pd R ðmþ < 1 and pd Rð R= mþ pd Rð mþþ1 < 1. By Lemma 1.2 w:gl:dim R=aR ¼ w:gl:dim R ¼ fd Rð R= mþ < 1: Therefore by [5] Corollary we have w:gl:dim R=aR ¼ w:gl:dim R 1: 2 CODIMENSION Theorem 2.1. Let R be a reglar indecomposable coherent semilocal ring. If every maximal ideal of R is finitely generated, then (1) m-codim R ðrþ ¼Codim Rm ðr m Þ¼w:gl:dim R m ; 8m 2 MaxðRÞ: (2) Codim R ðrþ ¼w:gl:dim R: Proof. (1) Sppose m is a maximal ideal of R. Since R is a coherent ring and m is finitely generated, it follows from [5] Theorem that R m is a

6 4816 TANG, HUANG, AND TONG coherent local ring with maximal ideal mr m and mr m is finitely generated, and w:gl:dim R m w:gl:dim R < 1. By Lemma 1.8, R m is reglar and mr m is generated by a t elements reglar R m -seqence and Codim Rm ðr m Þ¼ w:gl:dim R m ¼ t < 1. We will prove m-codim R ðrþ ¼t by indction on t. If t ¼ 0, then R m is a field and mr m ¼ 0. Since R is a reglar indecomposable coherent semilocal ring, it follows from [11] Corollary 5 that R is a GCD domain and ths m ¼ 0andm-codim R ðrþ ¼0. Now sppose t > 0. If m ¼ 0, then R is a field and t ¼ w:gl:dim R m ¼ 0, which is a contradiction. So m 6¼ 0. By Lemma 1.6, there exists an irredcible element a of R in m sch that R=aR is a local ring with maximal ideal m=ar and a, considered as an element in R m, can be extended to a minimal set of generators of mr m. By Lemma 1.9 we have w:gl:dim R m =ar m ¼ w:gl:dim R m 1 ¼ t 1: We denote R=aR by R, then R is also coherent. Since R is a local ring with maximal ideal m ¼ m=ar, R m R, and since it is easy to verify that R m R m =ar m ; w:gl:dim R ¼ w:gl:dim R m ¼ w:gl:dim R m =ar m ¼ w:gl:dim R m 1 ¼ t 1 < 1: By indction hypothesis, we have m-codim Rð RÞ ¼t 1. In addition, m-codim R ðrþ ¼ m-codim Rð RÞþ1, so m-codim R ðrþ ¼t. The statement (2) is an immediate conseqence of (1). Corollary 2.2. Let R be a reglar coherent semilocal ring. If every maximal ideal of R is finitely generated, then (1) m-codim R ðrþ ¼Codim Rm ðr m Þ¼w:gl:dim R m ; 8m 2 MaxðRÞ: (2) Codim R ðrþ ¼w:gl:dim R: Proof. By Lemma 1.7, R has an indecomposable decomposition: R ¼ R 1 R 2 R n, where every R i ð1 i nþ is a reglar indecomposable coherent semilocal ring. Now, 8m 2 MaxðRÞ we assme m ¼ m 1 R 2 R n ; m 1 2 MaxðR 1 Þ: It is easy to verify that m-codim R ðrþ ¼m 1 -codim R1 ðr 1 Þ. Then by Theorem 2.1 we have m-codim R ðrþ ¼m 1 -codim R1 ðr 1 Þ¼w:gl:dim R 1m2 ¼ w:gl:dim R m : The statement (2) is an immediate conseqence of (1).

7 CODIMENSION AND REGULARITY REGULARITY OF COHERENT SEMILOCAL RINGS Theorem 3.1. Let R be a coherent semilocal ring. If every maximal ideal of R is finitely generated, then the following statements are eqivalent: (1) R is reglar; (2) w:gl:dim R < 1; (3) R m is reglar, 8m 2 MaxðRÞ; (4) pd R ðr=mþ < 1; 8m 2 MaxðRÞ; (5) pd R ðr=jþ < 1; (6) id R ðr=mþ < 1, 8m 2 MaxðRÞ; (7) id R ðr=jþ < 1; (8) FP-id R ðr=mþ < 1, 8m 2 MaxðRÞ; (9) FP-id R ðr=jþ < 1. Proof. Assme that m 1 ;...; m t are all maximal ideals of R. Then R=J t i¼1 R=m i and w:gl:dim R ¼ spfw:gl:dim R mi j1 i tg¼ spf fd R ðr=m i Þj1 i tg: By Lemma 1.2 we have w:gl:dim R ¼ fd R ðr=jþ ¼id R ðr=jþ ¼FPid R ðr=jþ. Since every m i ð1 i tþ is finitely generated, J ¼ T t Q i¼1 m i ¼ t i¼1 m i is also finitely generated. Ths by Lemma 1.1 fd R ðr=m i Þ¼ pd R ðr=m i Þ, for any 1 i t and therefore fd R ðr=jþ ¼pd R ðr=jþ: Now we only need to prove ð2þ )ð1þ. The other implications are trivial by above argment. By [5] Corollary we have w:gl:dim R ¼fpd R ðr=iþ ji is a finitely generated ideal of Rj jpd R ðiþþ1 j I is a finitely generated ideal of Rg. Since w:gl:dim R < 1, pd R ðiþ < 1 for every finitely generated ideal I of R. Ths R is reglar. Theorem 3.2. Let R be an indecomposable coherent semilocal ring. If every maximal ideal of R is finitely generated, then the following statements are eqivalent: (1) R is reglar; (2) Every maximal ideal m of R is generated by a reglar R-seqence with q elements, where q ¼ w:gl:dim R m ; (3) Every maximal ideal m of R satisfying w:gl:dim R m ¼ w:gl:dim R is generated by a reglar R-seqence. (4) There is a maximal ideal m of R sch that w:gl:dim R ¼ w:gl:dim R m and m is generated by a reglar R-seqence. Proof. The implications of ð2þ )ð3þ )ð4þ are trivial. ð4þ )ð1þlet m 2 MaxðRÞ with w:gl:dim R ¼ w:gl:dim R m and let m be generated by a reglar R-seqence a 1 ;...; a t. Then by Lemma 1.3, a 1 ;...; a t,

8 4818 TANG, HUANG, AND TONG considered as elements in mr m, is a reglar R m -seqence in mr m and mr m ¼ða 1 ;...; a t ÞR m, and ths by Lemma 1.8 we have that w:gl:dim R m ¼ Codim Rm ðr m Þ¼t. Sow:gl:dim R ¼ w:gl:dim R m < 1 and hence R is reglar by Theorem 3.1. ð1þ )ð2þ Sppose m is any maximal ideal of R and w:gl:dim R m ¼ q. If q ¼ 0, the conclsion is trivial. Now sppose q > 0. It follows from [11] Corollary 5 that R is a GCD domain. By Lemma 1.6, there exists an irredcible element a 1 of R in m sch that R=a 1 R (denoted by R) is a local ring with niqe maximal ideal m ¼ m=a 1 R. Similar to the proof of Theorem 2.1, we can prove w:gl:dim R ¼ w:gl:dim R m 1 ¼ q 1 < 1.So R is a reglar coherent local ring whose niqe maximal ideal m is also finitely generated. It follows from Lemma 1.8 that m is generated by a reglar Rseqence: a 2 ;...; a q. Ths m ¼ða 1 ; a 2 ;...; a q Þ and a 1 ; a 2 ;...; a q is a reglar R-seqence. Corollary 3.3. Let R be a coherent semilocal ring and let R ¼ R 1 R 2 R t be an indecomposable decomposition of R. If every maximal ideal of R is finitely generated and w:gl:dim R > 0, then the following statements are eqivalent: (1) R is reglar; (2) Forevery maximal ideal m ¼ R 1 R i 1 m i R iþ1 R t, m i 2 MaxðR i Þ, of R, if m i 6¼ 0, then m is generated by a q elements reglar R-seqence, where q ¼ w:gl:dim R m ¼ w:gl:dim R imi. (3) Every maximal ideal m of R which satisfies w:gl:dim R ¼ w:gl:dim R m is generated by a reglar R-seqence. (4) There is a maximal ideal m of R sch that w:gl:dim R m ¼ w:gl:dim R and m is generated by a reglar R-seqence. Proof. ð2þ )ð3þ )ð4þ Clear. ð4þ )ð1þ It is similar to the proof of ð4þ )ð1þ in the proof of Theorem 3.2. ð1þ )ð2þ For convenience sake, we assme that m ¼ m 1 R 2 R t, m 1 2 MaxðR 1 Þ, m 1 6¼ 0. Since w:gl:dim R > 0, sch an m mst exist. Since w:gl:dim R ¼ spfw:gl:dim R i j1 i tg, w:gl:dim R 1 w:gl:dim R < 1,andthsR 1 is a reglar indecomposable coherent semilocal ring. Therefore by Theorem 3.2 the maximal ideal m 1 of R is generated by a q elements reglar R 1 -seqence a 0 1 ;...; a0 q. Set a i ¼ a 0 i þ e 2 þþe t, 1 i q, where e i ð2 i tþ is the identity of R i. It is easy to show that a 1 ;...; a q is a reglar R-seqence and m ¼ða 1 ;...; a q Þ.Somis generated by a q elements R-seqence and q ¼ w:gl:dim R 1m1 ¼ w:gl:dim R m.

9 CODIMENSION AND REGULARITY 4819 Remark. When R satisfies the conditions of Corollary 3.3 and R is reglar, we can not obtain that every maximal ideal of R is generated by a reglar R- seqence. For example, let ðr 1 ; m 1 Þ be a noetherian local ring and 0 < gl:dim R 1 < 1. Set R ¼ R 1 F, F is a field. It is easy to see that R satisfies the conditions of Corollary 3.3 and 0 < w:gl:dim R < 1. Bt the maximal ideal m ¼ R 1 0 6¼ 0 can not be generated by a reglar R-seqence. The Jacobson radical of R is J ¼ m 1 0 6¼ 0 and J also can not be generated by a reglar R-seqence. 4 SOME EXAMPLES In this section, we constrct some interesting examples of nonnoetherian reglar indecomposable coherent semilocal rings. Proposition 4.1. Let t; n be any two natral nmbers. Then there exists a non-noetherian indecomposable coherent semilocal ring R sch that: (1) R has exactly t maximal ideals; (2) w:gl:dim R ¼ n þ 1, gl:dim R ¼ n þ 2; (3) Every maximal ideal of R is not finitely generated. Proof. Let K be a field with characteristic zero. Set A ¼ S n1 K½½x1 n ŠŠ. It follows from [8] Example 2 that A is a non-noetherian coherent local ring with niqe maximal ideal m ¼ffðxÞ 2A j f ð0þ ¼0g which is not finitely generated and w:gl:dim A ¼ 1, gl:dim A ¼ 2. Let L ¼ A½t 1 ; t 2 ;...; t n Š be the polynomial ring over A on the indeterminates t 1 ; t 2 ;...; t n. Set m i ¼ m þðt 1 þ i 1ÞL þ t 2 L þþt n L; 1 i t: It is easy to verify that m 1 ;...; m t are different maximal ideals of L. Set S ¼ L S t i¼1 m i ¼fs2Ljs =2 S t i¼1 m ig. It is clear that S is a mltiplicatively closed sbset of L and 1 2 S, we denote by L S the localization ring of L at S. We shall prove that L S satisfies all conditions of Proposition 4.1. First, by [11] Corollary 5, A is a domain. So L and L S are domains and hence L S is indecomposable. Since A is a coherent ring of global dimension two, it follows from [5] Theorem that L is a coherent ring and ths from [5] Theorem we know that L S is also a coherent ring. It is easy to verify that m 1 L S ;...; m t L S are all different maximal ideals of L S and every m i L S ð1 i tþ is not finitely generated. By [5] Hilbert Syzygies Theorem , w:gl:dim L ¼ w:gl:dim A þ n ¼ n þ 1 and gl:dim L ¼ gl:dima þ n ¼ n þ 2. It follows from [5] Theorem that w:gl:dim L S w:gl:dim L ¼ n þ 1 and gl:dim L S gl:dim L ¼ n þ 2. Let 0 6¼ a 2 m, we may easily show

10 4820 TANG, HUANG, AND TONG that a; t 1 ; t 2 ;...; t n is a reglar L-seqence in m 1. By Lemma 1.3, a; t 1 ;...; t n, considered as elements in L m1, is a reglar R m1 -seqence and ths by Theorem 2.1 we have w:gl:dim L m1 ¼ Codim L m1 n þ 1. Set S 1 ¼ L m 1, then S S 1. It follows from [2] x 7.2 Proposition 7.4 that L m1 ¼ L S1 ðl S Þ S1 =S. Ths by [5] Theorem we have w:gl:diml S w:gl:diml m1 n þ 1 and gl:diml S gl:diml m1 n þ 1: Therefore w:gl:dim L S ¼ n þ 1andnþ1gl:dim L m1 gl:dim L S n þ 2: If gl:dim L m1 ¼ n þ 1, then gl:dim L m1 ¼ w:gl:dim L m1 ¼ n þ 1 < 1 and so by [5] Theorem the maximal ideal m 1 L m1 is finitely generated, which is a contradiction. Hence gl:dim L m1 ¼ n þ 2. In addition, gl:dim L m1 gl:dim L S n þ 2. Ths gl:dim L S ¼ n þ 2. Proposition 4.2. Let t; n be any two natral nmbers. Then there exists a non-noetherian indecomposable coherent semilocal ring R sch that: (1) R has exactly t maximal ideals; (2) w:gl:dim R ¼ gl:dim R ¼ n þ 2; (3) Every maximal ideal of R is finitely generated. Proof. Let ða; mþ be an mbrella ring (see [9] for the definition) with characteristic zero (For example, let L ¼ Z½xŠ, m ¼ð2; xþ2maxðlþ, T ¼ L m, K the qotient field of T. It follows from [9] that A ¼ K½½tÞÞ ¼ f f ðtþ 2K½½tŠŠ j f ð0þ 2Tg is an mbrella ring and it is easy to see that the characteristic of A is zero). By [8] Example 3, we know that A is a coherent local domain and w:gl:dim A ¼ gl:dim A ¼ 2 and m can be generated by a two elements reglar A-seqence fa 1 ; a 2 g and there is a non-finitely generated prime ideal of A.So A is a non-noetherian sper reglar coherent local ring. Set L ¼ A½t 1 ; t 2 ;...; t n Š; m i ¼ða 1 ; a 2 ; t 1 þ i 1; t 2 ;...; t n Þ; 1 i t: Since Char A ¼ 0, it is easy to prove that m 1 ;...; m t are different maximal ideals of L. Set ( ) S ¼ L [t m i ¼ s 2 Ljs =2 [t : i¼1 m i i¼1 We can prove that L S satisfies the conditions of Proposition 4.2. The proof here is similar to that of Proposition 4.1, we omit it.

11 CODIMENSION AND REGULARITY 4821 ACKNOWLEDGMENT Spported by the National Science Fondation of China ( ). REFERENCES 1. Aslander, M.; Bchsbam, D.A. Homological Dimension in Noetherian Rings II. Trans. Amer. Math. Soc. 1958, 88, Cartan, H.; Eilenberg, S. Homological Algebra; Princeton Univ. Press: Princeton, Cheng, F.C. Homological Algebra; Gangxi Normal Univ. Press: Gilin, Feng, K.Q. A First Corse to Commtative Algebra; Higher Edcation Press: Beijing, Glaz, S. Commtative Coherent Rings. Lectre Notes in Math. 1371; Berlin Heidelberg: Springer-Verlag, Hang, Z.Y. Homological Dimension Over Coherent Semilocal Rings II. Pitman Research Notes in Math. Series 1996, 346, Kaplansky, I. Commtative Rings; Univ. of Chicago Press: Chicago, Tang, G.H.; Zhao, M.Q.; Tong, W.T. On the Reglarity of Coherent Local Rings. Acta Mathematica Sinica, to appear. 9. Vasconcelos, W.V. The Local Rings of Global Dimension Two. Proc. Amer. Math. Soc. 1970, 35, X, J.Z. Modles and Homological Dimensions Over Semilocal Rings. Chinese Ann. Math. 1986, 7A (6), Zhao, Y.C. On Commtative Indecomposable Coherent Reglar Rings. Comm. in Alg. 1992, 20 (5), Received October 1999 Revised Jne 2000