# Strongly Clean Matrix Rings Over Commutative Rings - Mathematics > Rings and Algebras

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Abstract: A ring $R$ is called strongly clean if every element of $R$ is the sum of aunit and an idempotent that commute.By { m SRC} factorization, Borooah, Diesl, and Dorsey\cite{BDD051} completely determined when ${\mathbb M} nR$ over acommutative local ring $R$ is strongly clean.We generalize the notion of { m SRC} factorization to commutative rings,prove that commutative $n$-{ m SRC} rings $n\ge 2$ are precisely thecommutative local rings over which ${\mathbb M} nR$ is strongly clean, andcharacterize strong cleanness of matrices over commutative projective-freerings having { m ULP}. The strongly $\pi$-regular property hence, stronglyclean property of ${\mathbb M} nCX,{\mathbb C}$ with $X$ a { m P}-spacerelative to ${\mathbb C}$ is also obtained where $CX,{\mathbb C}$ is the ringof complex valued continuous functions.

Autor: Lingling Fan, Xiande Yang

Fuente: https://arxiv.org/