Finite rings in varieties with definable principal congruences
Author:
G. E. Simons
Journal:
Proc. Amer. Math. Soc. 121 (1994), 649655
MSC:
Primary 16R10; Secondary 08B26, 16P10, 16R40
MathSciNet review:
1207541
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Abstract 
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Additional Information
Abstract: A variety of rings has definable principal congruences if there is a firstorder formula defining principal twosided ideals for all rings in . Any variety of commutative rings has definable principal congruences, but many noncommutative 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 in a variety with definable principal congruences.
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 E. Kiss, Definable principal congruences in congruence distributive varieties, Algebra Universalis 21 (1985), 213224. MR 855740 (88c:08007)
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 R. McKenzie, Paraprimal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties, Algebra Universalis 8 (1978), 336348. MR 0469853 (57:9634)
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 , Residually small varieties of Kalgebras, Algebra Universalis 14 (1982), 181196. MR 634997 (83d:08012b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412075415
PII:
S 00029939(1994)12075415
Keywords:
Definable principal congruences,
varieties of rings,
finite rings,
subdirectly irreducible rings,
finite local rings
Article copyright:
© Copyright 1994
American Mathematical Society
