Strong cleanness of matrix rings over commutative rings

Communications in Algebra (Impact Factor: 0.39). 05/2008; 36(2). DOI: 10.1080/00927870701715175
Source: arXiv

ABSTRACT Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $\pi$-regular commutative ring.

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Available from: François Couchot, Apr 04, 2015
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    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. Aring 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.
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