Related papers: VIC-modules over noncommutative rings

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if…

In this note, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules $\left(M,N \right)$ with $M$ two-periodic, the…

Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only…

Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a chain complex of modules over the subring P of elements of non-negative degree. We show that there are non-commutative localisations of P which detect whether the…

Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $\mathcal{D}(R)$ denote the derived category of $R$-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let $X$ be a…

Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily…

For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$--artinian…

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the…

Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of…

We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is…

We give equivalences between given properties of a commutative ring, and other properties on its ring of Witt vectors. Amongst them, we characterise all commutative rings whose rings of Witt vectors are Noetherian. We define a new category…

We define, via Gorenstein homomorphisms, a class of local rings over which there exist non-trivial totally reflexive modules. We also provide a general construction of such rings, which indicates their abundance.

Let $Q$ be a finite acyclic quiver and $A_Q$ the cluster algebra of $Q$. It is well-known that for each field $k$, the additive equivalence classes of support tilting $kQ$-modules correspond bijectively with the clusters of $A_Q$. The aim…

In this paper we extend two results of Happel to commutative rings. Let $(A, \mathfrak{m})$ be a commutative Noetherian local ring. Let $D^b_f(mod \ A)$ be the bounded derived category of complexes of finitely generated modules over $A$…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

Let A be a commutative ring, and \a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM…

Let $R$ be commutative Noetherian ring and let $\fa$ be an ideal of $R$. For complexes $X$ and $Y$ of $R$--modules we investigate the invariant $\inf{\mathbf R}\Gamma_{\fa}({\mathbf R}\Hom_R(X,Y))$ in certain cases. It is shown that, for…

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective…

We combine a combinatorial idea of Benjamin Weiss and some localization theory of Grothendieck categories to give a short and completely self-contained proof of the following recent result of Hanfeng Li and Bingbing Liang: Given a left…