Characterizations of Quasipolar Rings

Author: Cui Jian  

Publisher: Taylor & Francis Ltd

ISSN: 0092-7872

Source: Communications in Algebra, Vol.41, Iss.9, 2013-09, pp. : 3207-3217

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Abstract

A ring R is quasipolar if for every a ∈ R there exists p2 = p ∈ R such that , a + p ∈ U(R) and ap ∈ Rqnil; the element p is called a spectral idempotent of a. Strongly π-regular rings are quasipolar and quasipolar rings are strongly clean. In this paper, the relationship among strongly regular rings, strongly π-regular rings and quasipolar rings are investigated. Moreover, we provide several equivalent characterizations on quasipolar elements in rings. Consequently, it is shown that any quasipolar element in a ring is strongly clean. It is also proved that for a module M, α ∈end(M) is quasipolar if and only if there exist strongly α-invariant submodules P and Q such that M = P ⊕ Q, α|P is isomorphic and α|Q is quasinilpotent, which is shown to be a natural generalization of Fitting endomorphisms.