Related papers: A note on simple modules over quasi-local rings

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module that is generically Gorenstein. In this paper, I prove isomorphisms relating the minimal MCM approximations and minimal FID hulls of modules constructed from a…

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the…

For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local…

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$…

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle…

Let $\bar{I}$ denote the integral closure of an ideal in a Noetherian ring $R$. The main result of this paper asserts that $R$ is locally quasi-unmixed if and only if, the topologies defined by $\overline{I^n}$ and $I^{\langle n\rangle}$,…

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length…

The class of semi-hereditary rings is an important class of rings in theories that do not assume the Noetherian condition, such as perfectoid ring theory. We prove several results concerning the structure theory of this class, focusing on…

Let R be a commutative ring, and let S be a multiplicative subset of R. In this paper, we introduce and investigate the notion of S-FP-injective modules. Among other results, we show that, under certain conditions, a ring R is S-Noetherian…

Let R be a commutative Noetherian d-dimensional complete equicharacterisitc regular local ring and let I be an ideal of R such that every minimal prime over I has height at most c. Let v=d - [(d-2)/c]-1 and v'=d - [(d-1)/c]. It has been…

The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then…

Let $R$ be a commutative ring with identity and $D$ an $R$-module. It is shown that if $D$ is pure injective, then $D$ is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows…

This article investigates the relationship between Betti numbers of finitely generated modules over a Noetherian local ring $(R, \mathfrak{m})$ and the structure of formal local cohomology modules. We establish a connection between the…

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem.…

Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~…

Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module $M$. The submodule $\langle E_M(N)\rangle$ generated by the envelope $E_M(N)$ of $N$ is instrumental in studying rings and modules that satisfy the radical…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is…

Let $A$ be a non Gorenstein Cohen Macaulay ring of dimension $d\geq 1$, $I$ an ideal of $A$, and suppose $\omega_A$ is a canonical $A$-module. Set $$r(I,\omega_A) = \bigcup_{n \geq 0} (I^{n+1} \omega_A : I^{n} \omega_A) \subseteq A .$$ We…

This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and…