Related papers: A new approach to Baer and dual Baer modules

We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is,…

For any finitely generated module $M$ with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant $\operatorname{h}(M)$ based on a partial trace ideal of $M$. We study its properties and…

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the…

Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but…

This is a survey paper, based on lectures given at the Workshop on "Structured ring spectra and their applications" which took place January 21-25, 2002, at the University of Glasgow. The term `Morita theory' is usually used for results…

Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$ into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the $n \times n$ determinant formula at these…

For rings R with identity, we define a class of nonlinear higher order recurrences on unitary left R-modules that include linear recurrences as special cases. We obtain conditions under which a recurrence of order k+1 in this class is…

Let $R$ by a right coherent ring and $R$-Mod denote the category of left $R$-modules. We show that there is an abelian model structure on $R$-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for…

In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is…

In this paper, we dualize the concept of {\Sigma}-Rickart modules as {\Sigma}-dual Rickart modules. An R-module M is said to be {\Sigma}-dual Rickart if the direct sum of arbitrary copies of M is dual Rickart. We prove that each…

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. It is shown that if $\Ext^i_R(R/\fa,M)$ is minimax for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is minimax for all $i\geq 0$ and for…

For a tensor ring $T_R(M)$, under certain conditions, we characterize the Gorenstein projective modules over $T_R(M)$, and prove that a $T_R(M)$-module $(X,u)$ is Gorenstein projective if and only if $u$ is monomorphic and ${\rm coker}(u)$…

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. In this article we answer affirmatively a question raised by the present author in \cite{B2}. Also, as an immediate consequence of this result it is shown that the…

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated $R$-module $M$ is Noetherian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with Ann$(M)\subseteq…

We study homological behavior of modules satisfying the Auslander condition. Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in…

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{\alpha }$ of $R$ indexed by the elements $\alpha \in G$ such that $R=\bigoplus_{\alpha \in G}R_{\alpha }$ and $R_{\alpha…

R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and…

A left and right noetherian semiperfect ring R is known to be indecomposable if and only if its factor by the second power of Jacobson radical is. This characterisation is used to study simple R-modules in terms of their Ext groups. It is…

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if any short exact sequence of the form $0\to K\to N\to M\to 0$ is neat-exact i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted to $N$. We…

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary…