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 {BDD05a} completely characterized the commutative local rings R for which Mₙ(R) is strongly clean. For a general local ring R and $n>1$, however, it is unknown when the matrix ring Mₙ(R) is strongly clean. Here we completely determine the local rings R for which M₂(R) is strongly clean.
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Yang et al. (2008) studied this question.