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

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

Citations:

Zbl 0946.16007
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References:

[1] DOI: 10.1016/j.jpaa.2007.05.020 · Zbl 1162.16016
[2] DOI: 10.1016/j.jalgebra.2005.08.005 · Zbl 1110.16029
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[11] DOI: 10.1016/j.jalgebra.2008.06.012 · Zbl 1162.16017
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