Related papers: On n-Perfect Rings and Cotorsion Dimension

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 study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative…

Let $A,B$ be two rings and let $ X$ be an $ A-$module. An additive map $h: A\to B$ is called n-ring homomorphism if $h(\Pi^n_{i=1}a_i)=\Pi^n_{i=1}h(a_i),$ for all $a_1,a_2, ...,a_n \in {A}$. An additive map $D: A\to X$ is called $n$-ring…

Let $M$ be a module. A {\em $\delta$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $\delta$-small kernel. A $\delta$-cover is said to be a {\em flat $\delta$-cover} in case $F$ is a flat module. In the present paper, we…

Inspired by a question raised by Eisenbud-Musta\c{t}\u{a}-Stillman regarding the injectivity of maps from ${\rm Ext}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings which we…

We construct Abelian model structures on the category of chain complexes over a ring $R$, from the notion homological dimensions of modules. Given an integer $n > 0$, we prove that the left modules over a ringoid $\mathfrak{R}$ with…

Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is…

Zhou defined $\delta$-semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal…

Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that of…

This paper introduces and studies homological properties of new classes of modules, namely, the $\mathcal F_1$-flat modules and the $\mathcal F_1^{\fp}$-flat modules, where $\mathcal F_1$ stands for the class of right modules of flat…

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel…

It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here…

We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness…

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…

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an…

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…

Let A be a noetherian commutative ring, and let I be an ideal in A. We study questions of flatness and I-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and I-adically free A-module.

In this paper we introduce the notion of "strong $n$-perfect rings" which is in some way a generalization of the notion of "$n$-perfect rings". We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a…

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a…