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 such that all possible structural matrix rings over 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 of a structural matrix ring over a semiprime Noetherian ring can be recovered, contrary to the fact that in general cannot be recovered.

**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