On Strongly Clean Matrix and Triangular Matrix Rings

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 Tn(R) is uniquely strongly clean for all n ≥ 1, if and only if R is abelian, Tn(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.

Strongly clean matrix rings over local rings

“…Examples of nonstrongly clean 2 × 2 matrices over a commutative local ring can be found in [10,11]. Recently, it was proved by Chen, Yang and Zhou in [4] that for each prime p, M 2 (Z (p) ) is not strongly clean, where Z (p) is the localization of Z at the prime ideal generated by p. In another recent paper [3], the same authors investigated when a 2 × 2 matrix ring M 2 (R) over a commutative local ring R is strongly clean, and they obtained a simple criterion for such a matrix ring to be strongly clean. However, their criterion cannot be used to determine whether an individual matrix A in M 2 (R) is strongly clean when the matrix ring M 2 (R) is not necessarily strongly clean.…”

A Survey of Strongly Clean Rings

“…For example, Camillo-Khurana [13] proved that unit regular rings are all clean, so Nicholson asked By [29], matrix rings over clean rings are still clean, but it is unknown if eRe is clean in case R is clean and e 2 = e ∈ R. In [37], Nicholson asked In her unpublished manuscript [40], Sánchez Campos proved that if R is strongly clean and e 2 = e ∈ R, then eRe is strongly clean. Later, two different proofs appeared in [17] and [20], respectively. Theorem 3.4 ([17, 20, 40]) Let R be a strongly clean ring.…”

“…The following example shows that the triangular matrix rings over local rings need not be strongly clean. We point out that T n (R) over a strongly π -regular ring R is strongly clean [17,20].…”

“…Several authors discussed when the matrix ring M n (R) over a commutative local ring R is strongly clean (see [10,17,18,22,28,32,43,44]). …”

Strongly P-Clean Rings and Matrices

“…But A ∈ M 2 (R) is strongly clean by [6,Corollary 2.2]. It is worth noting that every strongly P -clean 2 × 2 matrix over integral domains must be an idempotent by Theorem 4.4.…”