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 of rings has definable principal congruences if there is a first-order formula defining principal two-sided ideals for all rings in . 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 in a variety with definable principal congruences.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1207541-5

Keywords:
Definable principal congruences,
varieties of rings,
finite rings,
subdirectly irreducible rings,
finite local rings

Article copyright:
© Copyright 1994
American Mathematical Society