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Do isomorphic structural matrix rings have isomorphic graphs?

Author(s): S. Dascalescu; L. van Wyk
Journal: Proc. Amer. Math. Soc. 124 (1996), 1385-1391.
MSC (1991): Primary 16S50, 16P40, 16N60
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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.

References:

1.: A. W. Chatters, Non-isomorphic rings with isomorphic matrix rings, Proc. Edinburgh Math. Soc. 36 (1993), 339--348. MR 94b:16035
2.: S. P. Coelho, Automorphism groups of certain structural matrix rings, Comm. Algebra 22 (1994), 5567--5586. CMP 95:02.
3.: S. Jøndrup, The group of automorphisms of certain subalgebras of matrix algebras, J. Algebra 141 (1991), 106--114. MR 92h:16034
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6.: J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, New York, 1967.
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Additional Information:

S. Dascalescu
Affiliation: Facultatea de Matematica, University of Bucharest, Str. Academiei 14, R 70109, Bucharest 1, Romania

L. van Wyk
Affiliation: Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa
Address at time of publication: Department of Mathematics, Texas A & M University, College Station, Texas 77843-3368
Email: lvw@maties.sun.ac.za

DOI: 10.1090/S0002-9939-96-03172-3
PII: S 0002-9939(96)03172-3
Keywords: Structural matrix ring, semiprime Noetherian ring, Boolean matrix, graph
Received by editor(s): December 7, 1993
Received by editor(s) in revised form: November 3, 1994
Communicated by: Lance W. Small
Copyright of article: Copyright 1996, American Mathematical Society


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