Strongly clean matrix rings over local rings

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. In this paper, we determine when a 2×2 matrix A over a commutative local ring is strongly clean. Several equivalent criteria are given for such a matrix to be strongly clean. Consequently, we obtain several equivalent conditions for the 2×2 matrix ring over a commutative local ring to be strongly clean, extending a result of Chen, Yang, and Zhou.