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

The Schr\"{o}der-Bernstein Theorem for sets is well known. The question of whether two subisomorphic algebraic structures are isomorphic to each other, is of interest. An $R$-module $M$ is said to satisfy the Schr\"{o}der-Bernstein (or SB)…

In this note, we investigate the Baer splitting problem over commutative rings. In particular, we show that if a commutative ring $R$ is $\tau_q$-semisimple, then every Baer $R$-module is projective.

We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential…

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer…

For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a…

Both the classes of $R$-coneat injective modules and its superclass, pure Baer injective modules, are shown to be preenveloping. The former class is contained in another one, namely, self coneat injectives, i.e. modules $M$ for which every…

Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem…

We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod,…

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer…

Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for…

Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N \in \mathcal{P}_{w}^{\dagger_\infty}$ (see, \cite{FLQ}). In this paper, we introduce and study some properties of weak…

Following O'Meara's result [Journal of Algebra and Its Applications Vol~\textbf{13}, No. 8 (2014)], it follows that the block matrix $A=\begin{pmatrix} B & 0 0 & 0 \end{pmatrix} \in M_{n+r}(R)$, $B\in M_n(R)$, $r\ge 1$, over a von Neumann…

A module $M$ is {called} stable if it has no nonzero projective direct summand. For a ring $ R $, we study conditions under which $R$-modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule.…

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

In this paper, we define and study a particular case of von Neumann regular notion called a weak von Neumann regular ring. It shown that the polynomial ring $R[x]$ is weak von Neumann regular if and only if $R$ has exactly two idempotent…

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $\Omega_R$ is…

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 $M_R$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_R$ satisfies the condition $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$), if $ma=0$ implies $m\sigma(a)=0$ (respectively, $m\sigma(a)=0$…