Matrix equivalence and isomorphism of modules

It is well known that if A and B are n × m matrices over a ring R, then coker A ? coker B does not imply A and B are equivalent. An elementary proof is given that the implication does hold if 1 is in the stable range of R. Furthermore, for certain R (including commutative rings), if A is block diagonal and B is block upper triangular with the same diagonal blocks as A, then coker A ? coker B implies A and B are equivalent under a special equivalence. This extends results of Roth and Gustafson. As a corollary, a theorem on decomposition of modules is obtained.