CS matrix rings over local rings

Diagonal Matrix Reduction over Refinement Rings

  Abstract: A ring R is called a refinement ring if the monoid of finitely generated projective R- modules is refinement.  Let R be a commutative refinement ring and M, N, be two finitely generated projective R-nodules, then M~N  if and only if Mm ~Nm for all maximal ideal m of  R. A rectangular matrix A over R admits diagonal reduction if there exit invertible matrices p and Q such that PAQ is...

Strongly Clean Matrix Rings over Commutative Rings

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 ring...

Differential Polynomial Rings of Triangular Matrix Rings

Orthogonal Groups over Local Rings

In an earlier paper [S] we have determined the structure of the linear groups over a local ring. In this note we continue the study of the classical groups over a local ring with the investigation of the orthogonal groups. Our main result (cf. Theorem 6 below) is a complete description of the invariant subgroups of an orthogonal group of noncompact type (i.e., of index ^ 1) over a local ring L ...

Strong 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 ...