Characterizations of Quasipolar Rings

Abstract

A ring R is quasipolar if for every a ∈ R there exists p² = p ∈ R such that , a + p ∈ U(R) and ap ∈ R^qnil; 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.

Key Words:

• Quasipolar ring
• Spectral idempotent
• Strongly clean ring
• Strongly π-regular ring

2010 Mathematics Subject Classification:

• 16U99
• 16N99
• 15A09

ACKNOWLEDGMENT

The authors are highly grateful to the referee for correcting many errors and valuable suggestions. This research is supported by the NNSF of China (10971024, 11201064), the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110020), the NSF of Jiangsu Province (BK2010393), and the NSF of Anhui Educational Committee (KJ2010A126).

Notes

Communicated by V. A. Artamonov.