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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Finite rings in varieties with definable principal congruences


Author: G. E. Simons
Journal: Proc. Amer. Math. Soc. 121 (1994), 649-655
MSC: Primary 16R10; Secondary 08B26, 16P10, 16R40
MathSciNet review: 1207541
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Abstract: A variety $ \mathcal{V}$ of rings has definable principal congruences if there is a first-order formula defining principal two-sided ideals for all rings in $ \mathcal{V}$. Any variety of commutative rings has definable principal congruences, but many non-commutative rings cannot be in a variety with definable principal congruences. We show that a finite ring in a variety with definable principal congruences is a direct product of finite local rings. This result is used to describe the structure of all finite rings R with $ J{(R)^2} = 0$ in a variety with definable principal congruences.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1207541-5
PII: S 0002-9939(1994)1207541-5
Keywords: Definable principal congruences, varieties of rings, finite rings, subdirectly irreducible rings, finite local rings
Article copyright: © Copyright 1994 American Mathematical Society