Quick answer: no, they are not isomorphic.

Long answer: The two block upper triangular matrix rings under consideration differ in the arrangement of their diagonal and off-diagonal blocks. The first ring has Mn(R) and Mm(R) as its diagonal blocks, with Mn×m(R) as the off-diagonal block. In contrast, the second ring has Mm(R) and Mn(R) as its diagonal blocks, with Mm×n(R) as the off-diagonal block. While their overall structure appears similar, their internal properties differ significantly.

Matrix rings Mn(R) and Mm(R) are well-known to be non-isomorphic if n≠m, as they correspond to matrices of different dimensions over the commutative ring R. This dimensional difference reflects in their structure as R-modules, making it impossible to map Mn(R) to Mm(R) through an isomorphism.

The two block upper triangular matrix rings are not isomorphic when n≠m. This is because the diagonal blocks, which are non-isomorphic matrix rings, and the off-diagonal blocks, which are non-identical bimodules, prevent a bijective ring homomorphism from existing between the two structures. However, if n=m, the rings become isomorphic as their diagonal and off-diagonal components align perfectly.

Dear Marcos, Thanks for your answer. But if two block upper trinagular rings are (ring) isomorphic, I can not prove the coresponding diagonals are (ring) isomorphic.