On the ideal case of a conjecture of Auslander and Reiten

2003

Suppose that A is a semiprimary ring satisfying one of the two conditions: 1) its Yoneda ring is generated in finite degrees; 2) its Loewy length is less or equal than three. We prove that the global dimension of A is finite if, and only if, there is a m> 0 such that Ext n A(S, S) = 0, for all simple A-modules S and all n≥m In a recent paper, Skowronski, Smal ∅ and Zacharia ([8]) proved that a left Artinian ring A has finite global dimension if, and only if, every finitely generated indecomposable left A-module has either finite projective dimension or finite injective dimension. Then, they showed that one cannot replace ’indecomposable’ by ’simple ’ in that statement, by giving a counterexample of Loewy length 4. Finally they asked the following question: Question: Suppose A is a left Artinian ring such that, for each finitely (M, M) = 0 for n ≫ generated indecomposable left A-module M, one has Extn A 0 (i.e. there is a m = m(M)> 0 such that Extn A (M, M) = 0 for all n ≥ m). ...

Arxiv preprint math/0407270, 2004

Abstract. Let R be a commutative Noetherian ring. It is shown that R is Artinian if and only if every R-module is good, if and only if every R-module is representable. As a result, it follows that every nonzero submodule of any representable R-module is representable if and only if R is ...

2017

Let R be a commutative ring with 1 = 0, and let I be a proper ideal of R. Recall that 20

Journal of Algebra, 2006

Let R be a Noetherian commutative ring with unit 1 = 0, and let I be a regular proper ideal of R. The set P(I) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to I and to P(I) a numerical semigroup S(I); we have S(I) = N if and only if every element of P(I) is the integral closure of a power of the largest element K of P(I). If this holds, the ideal K and the set P(I) are said to be projectively full. A special case of the main result in this paper shows that if R contains the rational number field Q, then there exists a finite free integral extension ring A of R such that P(I A) is projectively full. If R is an integral domain, then the integral extension A has the property that P((I A + z *)/z *) is projectively full for all minimal prime ideals z * in A. Therefore in the case where R is an integral domain there exists a finite integral extension domain B = A/z * of R such that P(I B) is projectively full.

Journal of Algebra, 2002

Let A be a finitely generated module over a (Noetherian) local ring R M . We say that a nonzero submodule B of A is basically full in A if no minimal basis for B can be extended to a minimal basis of any submodule of A properly containing B. We prove that a basically full submodule of A is M-primary, and that the following properties of a nonzero M-primary submodule B of A are equivalent: (a) B is basically full in A;

2021

Let $(R, mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $mathrm{pd}_R, A=dim R$. Furthermore, as an application of these results it is shown that$$pdd H^i_{{frak p}R_{frak p}}(M_{frak p}, N_{frak p})leq pd H^{i+dim R/{frak p}}_{frak m}(M,N)$$for each ${frak p}in mathrm{Spec} R$ and each integer $igeq 0$. This result answers affirmatively a question raised by the present authors in [13].

Czechoslovak Mathematical Journal, 2013

Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0 : M p) is not finitely generated and for each i ) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N ) ⊆ V (I) and for all integers i 0, the R-modules Ext i R (N, M ) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N ) ⊆ V (I) and for all integers i 0, the R-modules Ext i R (M, N ) are of finite length.

Arxiv preprint math/0606694, 2006

This paper partly settles a conjecture of Costa on (n, d)-rings, i.e., rings in which n-presented modules have projective dimension at most d. For this purpose, a theorem studies the transfer of the (n, d)-property to trivial extensions of local rings by their residue fields. It concludes with a brief discussion -backed by original examples-of the scopes and limits of our results.

It is proved that a noetherian commutative local ring A containing a field is regular if there is a complex M of free A-modules with the following properties: M i = 0 for i / ∈ [0, dim A]; the homology of M has finite length; H 0 (M) contains the residue field of A as a direct summand. This result is an essential component in the proofs of the McKay correspondence in dimension 3 and of the statement that threefold flops induce equivalences of derived categories.

Proceedings of the Edinburgh Mathematical Society, 2003

It was shown by Huynh and Rizvi that a ring $R$ is semisimple artinian if and only if every continuous right $R$-module is injective. However, a characterization of rings, over which every finitely generated continuous right module is injective, has been left open. In this note we give a partial solution for this question. Namely, we show that for a right semi-artinian ring $R$, every finitely generated continuous right $R$-module is injective if and only if all simple right $R$-modules are injective.AMS 2000 Mathematics subject classification: Primary 16D50. Secondary 16P20; 16P60