PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 5, May 1996 DO ISOMORPHIC STRUCTURAL MATRIX RINGS HAVE ISOMORPHIC GRAPHS? S. D ˇ ASC ˇ ALESCU AND L. VAN WYK (Communicated by Lance W. Small) Abstract. We first provide an example of a ring R such that all possible 2 × 2 structural matrix rings over R are isomorphic. However, we prove that the underlying graphs of any two isomorphic structural matrix rings over a semiprime Noetherian ring are isomorphic, i.e. the underlying Boolean matrix B of a structural matrix ring M(B, R) over a semiprime Noetherian ring R can be recovered, contrary to the fact that in general R cannot be recovered. 1. Introduction and preliminaries Various authors have studied the relationship between two rings R and S for which the full matrix rings M n (R) and M n (S) are isomorphic. For example, Chat- ters [1] showed that one is very far from recovering the ring R from M n (R), even in the prime Noetherian case: he provided an uncountable family of pairwise non- isomorphic such rings with isomorphic full 2 × 2 matrix rings. Certain subrings of M n (R), for example tiled matrix rings, have also recently received considerable attention (see [4]). A particular class of tiled matrix rings are the structural matrix rings, having the tiles 0 or R. Instances of such rings have been intensively studied. For example, the automorphisms of structural matrix rings over certain rings were described in [2] and [3], and a left Artinian CI-prime ring was characterised as a complete blocked triangular matrix ring over a division ring in [5]. The properties of a structural matrix ring M(B,R), being a generalisation of both a full matrix ring and an upper triangular matrix ring, are influenced by the underlying Boolean matrix B (or, equivalently, the underlying graph) and the un- derlying ring R. We noted in the first paragraph that it is not in general possible to recover the ring R from M n (R). However, it might be possible to recover the Boolean matrix B from the structural matrix ring M(B,R). There are of course trivial examples of isomorphic structural matrix rings over a ring R having Boolean matrices of different orders, viz. consider any ring R for which R ∼ = R × R. More- over, we provide an example of a ring R for which all the 2 × 2 structural matrix rings (i.e. the Boolean matrices have the same order) are isomorphic. However, for certain rings this phenomenon cannot occur. The main result of this paper states Received by the editors December 7, 1993 and, in revised form, November 3, 1994. 1991 Mathematics Subject Classification. Primary 16S50, 16P40, 16N60. Key words and phrases. Structural matrix ring, semiprime Noetherian ring, Boolean matrix, graph. c 1996 American Mathematical Society 1385 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use