Some decompositions of matrices over local rings

Let R be a local ring with maximal ideal m, let n be a natural number greater than 1 and let A = (a(ij)) be a matrix in the general linear group GL(n)(R) of degree n over R. We firstly show that if the matrix ((a(ij)) over bar) is nonscalar in GL(n)(R/m) and s(1), s(2), . . . , s(n-1) are invertible elements in R, then there exists an invertible element s(n) is an element of R such that A is similar to the product VU in which V is a lower uni-triangular matrix and U is an upper triangular matrix whose diagonal entries are s(1), s(2), . . . , s(n). We then present some applications of this factorization to find decompositions of matrices in GL(n)(R) into product of commutators and involutions.