A note on strongly clean matrix rings. (English) Zbl 1191.16029

Fan, Lingling; Yang, Xiande

Commun. Algebra 38, No. 3, 799-806 (2010).

Summary: Let \(R\) be an associative ring with identity. An element \(a\in R\) is called strongly clean if \(a=e+u\) with \(e^2=e\in R\), \(u\) a unit of \(R\), and \(eu=ue\). A ring \(R\) is called strongly clean if every element of \(R\) is strongly clean. Strongly clean rings were introduced by W. K. Nicholson [Commun. Algebra 27, No. 8, 3583-3592 (1999; Zbl 0946.16007)]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when \(R\) is local or strongly \(\pi\)-regular.
In this note, necessary conditions for the matrix ring \(\mathbb{M}_n(R)\) (\(n>1\)) over an arbitrary ring \(R\) to be strongly clean are given, and the strongly clean property of \(\mathbb{M}_2(RC_2)\) over the group ring \(RC_2\) with \(R\) local is obtained.

MSC:

16S50	Endomorphism rings; matrix rings
16U60	Units, groups of units (associative rings and algebras)
16L30	Noncommutative local and semilocal rings, perfect rings
16S34	Group rings

Keywords:

group rings; local rings; matrix rings; strongly clean rings; idempotents; units

Citations:

Zbl 0946.16007

References:

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