Uniquely Strongly Clean Triangular Matrix Rings

2013/07/28 by H. Chen, Chen, H., O. Gurgun +3
Mathematics · #16S99 #16U99 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16S99 #msc:16U99

paper · pdf · doi:10.48550/arxiv.1307.7339

arxiv created 2013/08/29 · arxiv updated 2013/08/30

Abstract

A ring R is uniquely (strongly) clean provided that for any a∈ R there exists a unique idempotent e∈ R (∈ comm(a)) such that a-e∈ U(R). Let R be a uniquely bleached ring. We prove, in this note, that R is uniquely clean if and only if R is abelian, and T_n(R) is uniquely strongly clean for all n≥ 1, if and only if R is abelian, T_n(R) is uniquely strongly clean for some n≥ 1. In the commutative case, the more explicit results are obtained. These also generalize the main theorems in [6] and [7], and provide many new class of such rings.

Related

• On Perfectly Clean Rings
• Very clean matrices over local rings
• Strongly clean triangular matrix rings with endomorphisms
• Strong cleanness of the 2× 2 matrix ring over a general local ring
• Rings whose Nil-Clean and Clean Elements are Uniquely Nil-Clean