## 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

Zbl 0946.16007
Full Text:

### 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 [3] Chen W., Comm. Algebra 34 pp 2374– (2006) [4] Dorsey , T. J. ( 2006 ). Cleanness and Strong Cleanness of Rings of Matrices. Ph.D thesis, University of California, Berkeley . [5] DOI: 10.1016/j.jalgebra.2006.10.032 · Zbl 1117.16017 [6] DOI: 10.4153/CMB-1972-025-1 · Zbl 0235.16008 [7] DOI: 10.1080/00927879908826649 · Zbl 0946.16007 [8] Sánchez Campos , E. ( 2002 ). On strongly clean rings. Unpublished . [9] DOI: 10.1017/S0004972700034493 · Zbl 1069.16035 [10] DOI: 10.1016/j.laa.2007.03.012 · Zbl 1122.16027 [11] DOI: 10.1016/j.jalgebra.2008.06.012 · Zbl 1162.16017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.