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Do isomorphic structural matrix rings
have isomorphic graphs?


Authors: S. Dascalescu and L. van Wyk
Journal: Proc. Amer. Math. Soc. 124 (1996), 1385-1391
MSC (1991): Primary 16S50, 16P40, 16N60
DOI: https://doi.org/10.1090/S0002-9939-96-03172-3
MathSciNet review: 1307508
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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
  • 4. L. S. Levy, J. C. Robson, and J. T. Stafford, Hidden matrices, Proc. London Math. Soc. 69 (1994), 277--308. MR 95d:16026
  • 5. M. S. Li and J. M. Zelmanowitz, Artinian rings with restricted primeness conditions, J. Algebra 124 (1989), 139--148. MR 90g:16006
  • 6. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, New York, 1967.
  • 7. L. van Wyk, Special radicals in structural matrix rings, Comm. Algebra 16 (1988), 421--435. MR 89a:16009

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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: https://doi.org/10.1090/S0002-9939-96-03172-3
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
Article copyright: © Copyright 1996 American Mathematical Society

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