Annihilator conditions related to the quasi-Baer condition

Year 2016, Volume: 45 Issue: 1, 95 - 105, 01.02.2016

A. Taherifar

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 $r(I) = eR$. The class EGE includes
quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete
set of orthogonal primitive idempotents (hence all Noetherian rings) and is
closed under direct product, full and upper triangular matrix rings, polynomial
extensions (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 $S \subseteq S_{r}(R)$ such that $r(I) =
r(RSR)$. 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 (resp.,
AE)-ring if and only if the closure of any union of clopen subsets of $Spec(R)$
is open (resp., $Spec(R)$ is an EZ-space).

Keywords

Quasi-Baer ring, AE-ring, EGE-ring, Spec(R), Semicentral idempotent, EZ-space

References

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