Topic
Torsionless module
About: Torsionless module is a research topic. Over the lifetime, 12 publications have been published within this topic receiving 210 citations. The topic is also known as: torsionless.
Papers
TL;DR: In this article, it was shown that a 6-dimensional algebra with a semi-Gorenstein-projective module M which is not torsionless is left weakly Gorenstein.
Abstract: Let A be an artin algebra. An A -module M will be said to be semi-Gorenstein-projective provided that Ext i ( M , A ) = 0 for all i ≥ 1 . All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on A such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then A is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra Λ with a semi-Gorenstein-projective module M which is not torsionless (thus not Gorenstein-projective). Actually, also the Λ -dual module M ∗ is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Sega. Since all the syzygy-modules of M and M ∗ are 3-dimensional, the example can be checked (and visualized) quite easily.
37 citations
TL;DR: In this article, the classical density theorem for rings with minimal left ideals is generalized to include semiprime rings possessing faithful finite dimensional left ideals, and the method of proof covers the infinite dimensional case as well.
18 citations
Posted Content•
TL;DR: For a commutative Artinian ring with radical square zero, a finitely generated torsionless module is projective if and only if it is self-orthogonal.
Abstract: In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\Lambda$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\Lambda$-module $M$ is projective if ${\rm Ext^1_\Lambda}(M,M)=0$. For a commutative Artinian ring $\Lambda$, a finitely generated torsionless $\Lambda$-module $M$ is projective if the following conditions are satisfied: (1) ${\rm Ext}^i_{\Lambda}(M,\Lambda)=0$ for $i=1,2,3$; and (2) ${\rm Ext}^i_{\Lambda}(M,M)=0$ for $i=1,2$. As a consequence of this result, we have that for a commutative Artinian ring $\Lambda$, a finitely generated Gorenstein projective $\Lambda$-module is projective if and only if it is selforthogonal.
16 citations
TL;DR: In this article, it was shown that if A is self-injective and all modules are reflexive, then all the simple modules are re-referredivative.
Abstract: Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are re...
2 citations
1 Jun 1998
TL;DR: In this paper, the authors introduce the notion of the Koszul complex, which holds an intermediate position between arithmetic and homological algebra, and describe the fundamental numerical invariants of a Noetherian local ring.
Abstract: After dimension, depth is the most fundamental numerical invariant of a Noetherian local ring R or a finite R -module M . While depth is defined in terms of regular sequences, it can be measured by the (non-)vanishing of certain Ext modules. This connection opens commutative algebra to the application of homological methods. Depth is connected with projective dimension and several notions of linear algebra over Noetherian rings. Equally important is the description of depth (and its global relative grade) in terms of the Koszul complex which, in a sense, holds an intermediate position between arithmetic and homological algebra. This introductory chapter also contains a section on graded rings and modules. These allow a decomposition of their elements into homogeneous components and therefore have a more accessible structure than rings and modules in general. Regular sequences Let M be a module over a ring R . We say that x ∈ R is an M-regular element if xz = 0 for z ∈ M implies z = 0, in other words, if x is not a zero-divisor on M . Regular sequences are composed of successively regular elements: Definition 1.1.1 A sequence x = x 1 ,…, x n of elements of R is called an M-regular sequence or simply an M-sequence if the following conditions are satisfied: (i) x i is an M /( x 1 , …, x i −1 ) M -regular element for i = 1,…, n , and (ii) M / xM ≠ 0. In this situation we shall sometimes say that M is an x-regular module . A regular sequence is an R -sequence. A weak M-sequence is only required to satisfy condition (i). Very often R will be a local ring with maximal ideal m, and M ≠ 0 a finite R -module.
1 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
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| 2021 | 1 |
| 2020 | 1 |
| 2015 | 1 |
| 2012 | 1 |
| 2009 | 1 |
| 2008 | 1 |