Bulletin of the
Korean Mathematical Society BKMS

An element of a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute, and $R$ is called strongly clean if every element of $R$ is strongly clean. Let $R$ be a noncommutative local ring, a criterion in terms of solvability of a simple quadratic equation in $R$ is obtained for $M_2(R)$ to be strongly clean.

Keywords: strongly clean ring, matrix ring, local ring, similarity

MSC numbers: 16U99, 16S50