Do isomorphic structural matrix rings have isomorphic graphs?

Dascalescu, S.; van Wyk, L.

doi:10.1090/S0002-9939-96-03172-3

Do isomorphic structural matrix rings have isomorphic graphs?
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by S. Dascalescu and L. van Wyk PDF

Proc. Amer. Math. Soc. 124 (1996), 1385-1391 Request permission

Abstract:

We first provide an example of a ring $R$ such that all possible $2\times 2$ 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 $\mathbb M(B,R)$ over a semiprime Noetherian ring $R$ can be recovered, contrary to the fact that in general $R$ cannot be recovered.

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