Dual π\pi-Rickart Modules

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S =$ End$_R(M)$. In this paper we introduce dual $\pi$-Rickart modules as a generalization of $\pi$-regular rings as well as that of dual Rickart modules. The module $M$ is called {\it dual $\pi$-Rickart} if for any $f\in S$, there exist $e^2=e\in S$ and a positive integer $n$ such that Im$f^n=eM$. We prove that some results of dual Rickart modules can be extended to dual $\pi$-Rickart modules for this general settings. We investigate relations between a dual $\pi$-Rickart module and its endomorphism ring.Comment: arXiv admin note: text overlap with arXiv:1204.234

On Abelian rings

Abstract Let α be an endomorphism of an arbitrary ring R with identity. In this note, we introduce the notion of α -abelian rings which generalizes abelian rings. We prove that α -reduced rings, α -symmetric rings, α -semicommutative rings and α -Armendariz rings are α -abelian. For a right principally projective ring R , we also prove that R is α -reduced if and only if R is α -symmetric if and only if R is α -semicommutative if and only if R is α -Armendariz if and only if R is α -Armendariz of power series type if and only if R is α -abelian

A generalization of reduced rings

Let R be a ring with identity. We introduce a class of rings which is a generalization of reduced rings. A ring R is called central rigidif for any a,b ∈ R, $a^2 b = 0$ implies ab belongs to the center of R.Since every reduced ring is central rigid, we study sufficient conditions for central rigid rings to be reduced. We prove that some resu lts of reduced rings can be extended to central rigid rings for this general setting, in particular, it is shown that every reduced ring is central rigid, every central rigid ring is central reversible, central sem icommutative, 2-primal, abelian and so directly finite.Let R be a ring with identity. We introduce a class of rings which is a generalization of reduced rings. A ring R is called central rigidif for any a,b ∈ R, $a^2 b = 0$ implies ab belongs to the center of R.Since every reduced ring is central rigid, we study sufficient conditions for central rigid rings to be reduced. We prove that some resu lts of reduced rings can be extended to central rigid rings for this general setting, in particular, it is shown that every reduced ring is central rigid, every central rigid ring is central reversible, central sem icommutative, 2-primal, abelian and so directly finite

On generalized principally quasi-Baer modules

Let R be an associative ring with identity. A right R-module M is called generalized principally quasi-Baer if for any m ∈ M , r R (m R) is left sunital as an ideal of R and the ring R is said to be right (left) generalized principally quasi-Baer if R is a generalized principally quasi-Baer right (left) R-module. In this paper, we investigate properties of generalized principally quasi-Baer modules and right (left) generalized principally quasi-Baer rings. Keywords: generalized principally quasi-Baer modules, right (left) generalized principally quasi-Baer rings, Sea R un anillo asociativo con identidad. Se dice que un módulo derecho M de tipo R es de tipo generalizado principalmente de tipo cuasi-Baer si para cualquier m ∈ M , r R (m R) es unitario de tipo s a la izquierda como un ideal de R y el anillo R se dice de tipo generalizado principalmente de tipo cuasi-Baer derecho (izquierdo) si R es un módulo generalizado principalmente de tipo cuasi-Baer derecho (izquierdo) de tipo R. En este artículo se investigan las propiedades de los módulos generalizados principalmente de tipo cuasi-Baer y los anillos derechos (izquierdos) generalizados principalmente de tipo cuasi-Baer. Palabras claves: módulos generalizados principalmente de tipo cuasi-Baer, anillos derechos (izquierdos) generalizados principalmente de tipo cuasi-Baer. MSC: 13C99, 16D80, 16U80