Strongly Clean Matrix Rings over Commutative Rings
نویسندگان
چکیده
A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative n-SRC rings (n ≥ 2) are precisely the commutative local rings over which Mn(R) is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having ULP. The strongly π-regular property (hence, strongly clean property) of Mn(C(X,C)) with X a P-space relative to C is also obtained where C(X,C) is the ring of complex valued continuous functions.
منابع مشابه
Some classes of strongly clean rings
A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.
متن کاملStrongly nil-clean corner rings
We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings, then $R/J(R)$ is nil-clean. In particular, under certain additional circumstances, $R$ is also nil-clean. These results somewhat improves on achievements due to Diesl in J. Algebra (2013) and to Koc{s}an-Wang-Zhou in J. Pure Appl. Algebra (2016). ...
متن کاملStrongly clean triangular matrix rings with endomorphisms
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,sigma)$ be the skew triangular matrix ring over a local ring $R$ where $sigma$ is an endomorphism of $R$. We show that $T_2(R,sigma)$ is strongly clean if and only if for any $ain 1+J(R), bin J(R)$, $l_a-r_{sigma(b)}: Rto R$ is surjective. Furt...
متن کاملStrong cleanness of matrix rings over commutative rings
Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is ...
متن کاملStrong Cleanness of the 2× 2 Matrix Ring over a General Local Ring
A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [3] completely characterized the commutative local rings R for which Mn(R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring Mn(R) is strongly clean. Here we completely determi...
متن کامل