Related papers: The derived category of a locally complete interse…

In this paper we prove that if $R$ is a commutative, reduced, local ring, then $R$ is Hopfian if and only if the ring $R[x]$ is Hopfian. This answers a question of Varadarajan, in the case when $R$ is a reduced local ring. We provide…

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and…

We describe a method for solving linear systems over the localization of a commutative ring $R$ at a multiplicatively closed subset $S$ that works under the following hypotheses: the ring $R$ is coherent, i.e., we can compute finite…

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each…

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm…

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…

For a Noetherian commutative ring $R$, let $H^i_I(R)$ be the $ i$-th local cohomology module of $R$ with respect to $I$. In \cite{Hel-08}, Hellus posed the question of identifying rings $R$ such that $\operatorname{injdim}_R…

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and…

We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of…

In this paper several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property $\Sigma$-semi-compact for modules and we characterize the…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

Let $R$ be a local ring and $M$ a finitely generated $R$-module. The complete intersection dimension of $M$--defined by Avramov, Gasharov and Peeva, and denoted $\cidim_R(M)$--is a homological invariant whose finiteness implies that $M$ is…

Let $R$ be a commutative noetherian ring with a dualizing complex. By recent work of Iyengar and Krause, the difference between the category of acyclic complexes and its subcategory of totally acyclic complexes measures how far $R$ is from…

It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N, then the Ext groups between M and N vanish from some step if and only if the Ext groups between N…

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring $(R,m)$ with $m^3=0$. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes,…

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its…

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the…

For a finitely generated module $M$, over a commutative Noetherian local ring $(R, \mathfrak{m})$, it is shown that there exist only a finite number of non--isomorphic top local cohomology modules $\mathrm{H}_{\mathfrak{a}}^{\mathrm{dim}…

We study the inverse problem for the versal deformation rings $R(\Gamma,V)$ of finite dimensional representations $V$ of a finite group $\Gamma$ over a field $k$ of positive characteristic $p$. This problem is to determine which complete…

In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of…