2008/03/14 by Lingling Fan, Fan, Lingling, Xiande Yang +1
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.0803.2176
arxiv created 2008/08/20 · arxiv updated 2009/12/01
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 \citeBDD051 completely determined when \mathbb Mn(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≥ 2) are precisely the commutative local rings over which \mathbb Mn(R) is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having \rm ULP. The strongly π-regular property (hence, strongly clean property) of \mathbb Mn(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.