DO ISOMORPHIC STRUCTURAL MATRIX-RINGS HAVE ISOMORPHIC GRAPHS

We first provide an example of a ring R such that all possible 2 x 2 s tructural matrix rings over R are isomorphic. However, we prove that t he underlying graphs of any two isomorphic structural matrix rings ove r a semiprime Noetherian ring are isomorphic, i.e. the underlying Bool ean matrix B of a structural matrix ring M(B, R) over a semiprime Noet herian ring R can be recovered,contrary to the fact that in general R cannot be recovered.