Strongly Clean Matrix Rings Over Commutative Rings

Abstract

A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\rm SRC} factorization, Borooah, Diesl, and Dorsey \cite{BDD051} completely determined when ${\mathbb{M}}_n(R)$ over a commutative local ring $R$ is strongly clean. We generalize the notion of {\rm SRC} factorization to commutative rings, prove that commutative $n$-{\rm SRC} rings $(n\ge 2)$ are precisely the commutative local rings over which ${\mathbb{M}}_n(R)$ is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having {\rm ULP}. The strongly $\ddot{I}€$-regular property (hence, strongly clean property) of ${\mathbb{M}}_n(C(X,\mathbb{C}))$ with $X$ a {\rm P}-space relative to $\mathbb{C}$ is also obtained where $C(X,\mathbb{C})$ is the ring of complex valued continuous functions.