arXiv subjects
Publications and source records attributed to Sait Halıcıoglu.
An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly $J$-clean. In this paper, we investigate strong $J$-cleanness of $M_2(R;s)$ for a local ring $R$ and $s\in R$. We determine the conditions under which elements of $M_2(R;s)$ are strongly $J$-clean.
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 $π$-Rickart modules as a generalization of $π$-regular rings as well as that of dual Rickart modules. The module $M$ is called {\it dual $π$-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 $π$-Rickart modules for this general settings. We investigate relations between a dual $π$-Rickart module and its endomorphism ring.
Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. In this paper we introduce $π$-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The module $M$ is called {\it $π$-Rickart} if for any $f\in S$, there exist $e^2=e\in S$ and a positive integer $n$ such that $r_M(f^n)=eM$. We prove that several results of Rickart modules can be extended to $π$-Rickart modules for this general settings, and investigate relations between a $π$-Rickart module and its endomorphism ring.