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Let R be an associative ring with identity. An element xR is said to be weakly exchange if there exists an idempotent eR such that exR and 1−e∊(1−x)R or 1−e∊(1+x)R. The ring R is said to be weakly exchange if all of its elements are weakly exchange. In this paper an element-wise characterization is given, and it is shown that weakly-Abel weakly exchange rings are weakly clean. Moreover, a relation between unit regular rings and weakly clean rings is also obtained.

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A ring R is called NLI (rings whose nilpotent elements form a Lie ideal) if for each aN(R) and bR, abbaN(R). Clearly, NI rings are NLI. In this note, many properties of NLI rings are studied. The main results we obtain are the following: (1) NLI rings are directly finite and left min-abel; (2) If R is a NLI ring, then (a) R is a strongly regular ring if and only if R is a Von Neumann regular ring; (b) R is (weakly) exchange if and only if R is (weakly) clean; (c) R is a reduced ring if and only if R is a n-regular ring; (3) If R is a NLI left MC2 ring whose singular simple left modules are Wnil-injective, then R is reduced.

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Acta Mathematica Hungarica
Cai Chuanren
Fang Hongjing
, and
Wei Junchao
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