Varieties of rings with definable principal congruences

Author:
G. E. Simons

Journal:
Proc. Amer. Math. Soc. **87** (1983), 397-402

MSC:
Primary 16A38; Secondary 08B05, 16A12, 16A70

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684626-6

MathSciNet review:
684626

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Abstract: A variety of rings has definable principal congruences (DPC) if there is a first order sentence defining principal two-sided ideals for all rings in . The key result is that for any ring , does not have DPC if . This allows us to show that if has DPC, then is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring , has DPC if and only if is commutative. An example of a finite, local, noncommutative ring with having DPC is given.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684626-6

Keywords:
Varieties of rings,
definable principal congruences,
polynomial identity rings

Article copyright:
© Copyright 1983
American Mathematical Society