DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

In this paper, we establish several necessary and sufficient conditions under which every regular matrix admits a diagonal reduction. We prove that every regular matrix over an exchange ring R admits diagonal reduction if and only if for any m, n is an element of N (m >= n + 1) and any regular X is an element of M-mxn(R), (X 0(mx(m-n))) is an element of M-m(R) is unit-regular if and only if for any m, n is an element of N (m >= n+1) and any regular X is an element of M-mxn(R), there exist an idempotent E is an element of M-m(R) and a completed U is an element of M-mxn (R) such that X = EU if and only if for any idempotents e is an element of R, f is an element of M-2(R), phi : eR congruent to f (2R) implies that there exists a completed u is an element of R-2 such that phi(e) = ue = fu. These shows that diagonal reduction over exchange rings behaves like stable ranges.