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Turkish Journal of Mathematics

ORHAN GÜRGÜN

DOI

10.3906/mat-1410-35

Abstract

An associative ring with identity is called quasipolar provided that for each $a\in R$ there exists an idempotent $p\in R$ such that $p\in comm^2(a)$, $a+p\in U(R)$ and $ap\in R^{qnil}$. In this article, we introduce the notion of quasipolar general rings (with or without identity). Some properties of quasipolar general rings are investigated. We prove that a general ring $I$ is quasipolar if and only if every element $a\in I$ can be written in the form $a=s+q$ where $s$ is strongly regular, $s\in comm^2(a)$, $q$ is quasinilpotent, and $sq=qs=0$. It is shown that every ideal of a quasipolar general ring is quasipolar. Particularly, we show that $R$ is pseudopolar if and only if $R$ is strongly $\pi$-rad clean and quasipolar.

Keywords

Quasipolar general rings, strongly clean general rings, strongly $\pi$-regular general rings, (generalized) Drazin inverse, pseudopolar rings

First Page

15

Last Page

26

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