Strong cleanness of the $2\times 2$ matrix ring over a general local ring

Details Strong cleanness of the $2\times 2$ matrix ring over a general local ring Strong cleanness of the $2\times 2$ matrix ring over a general local ring

2008-05-03

arxiv.org/abs/0805.0359v1

Mathematics

Rings and Algebras

12 pages, to appear in Journal of Algebra

Scientific paper

A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.

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