Annihilator conditions related to the quasi-Baer condition

Authors : A. TAHERİFAR

Pages : 95-105

View : 15 | Download : 6

Publication Date : 2016-02-01

Article Type : Research Paper

Abstract :We call a ring R an EGE-ring if for each $I \leq R$, which is generated by a subset of right semicentral idempotents there exists an idempotent $e$ such that $rinsert ignore into journalissuearticles values(I); = eR$. The class EGE includes quasi-Baer, semiperfect rings insert ignore into journalissuearticles values(hence all local rings); and rings with a complete set of orthogonal primitive idempotents insert ignore into journalissuearticles values(hence all Noetherian rings); and is closed under direct product, full and upper triangular matrix rings, polynomial extensions insert ignore into journalissuearticles values(including formal power series, Laurent polynomials, and Laurent series); and is Morita invariant. Also we call $R$ an AE-ring if for each $I \unlhd R$, there exists a subset $S \subseteq S_{r}insert ignore into journalissuearticles values(R);$ such that $rinsert ignore into journalissuearticles values(I); = rinsert ignore into journalissuearticles values(RSR);$. The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring $R$, it is shown that $R$ is an EGE insert ignore into journalissuearticles values(resp., AE);-ring if and only if the closure of any union of clopen subsets of $Specinsert ignore into journalissuearticles values(R);$ is open insert ignore into journalissuearticles values(resp., $Specinsert ignore into journalissuearticles values(R);$ is an EZ-space);.
Keywords : Quasi Baer ring, AE ring, EGE ring, Spec R, , Semicentral idempotent, EZ space