A Survey of Strongly Clean Rings

Abstract

Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu = ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181-198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.

On Strongly Clean Matrix and Triangular Matrix Rings

“…In general, matrix rings have not such properties (cf. [10,Proposition 11.8]). Thus, it is attractive to investigate uniquely strongly cleanness of triangular matrix rings.…”

“…[6, Question 12] and [10,Question 11.13] asked that if "commutative" in the preceding result can be replaced by "abelian". The motivation of this note article is to explore this problem.…”

Characterizations of special clean elements and applications

“…On the other hand, uniquely (strongly) clean rings have notably been studied in [2], [8], [9], [7], [30], [38], and uniquely clean elements by Khuruna et al [23]. It is notably known that a ring is uniquely clean iff it is uniquely strongly clean and abelian, but that there exists uniquely strongly clean rings that are not abelian (hence not uniquely clean).…”

Uniquely strongly clean triangular matrices

“…[5]). In general, matrix rings do not have such properties (see [13,Proposition 11.8]). Thus, it is attractive to investigate uniquely strong cleanness of triangular matrices over a ring.…”

“…[5, Question 12] and [13,Question 11.13] asked if "commutative" in the preceding result can be replaced by "abelian". The motivation of this note is to explore this problem.…”