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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 $ \mathcal{V}$ of rings has definable principal congruences (DPC) if there is a first order sentence defining principal two-sided ideals for all rings in $ \mathcal{V}$. The key result is that for any ring $ R$, $ V({M_n}(R))$ does not have DPC if $ n \geqslant 2$. This allows us to show that if $ V(R)$ has DPC, then $ R$ is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring $ R$, $ V(R)$ has DPC if and only if $ R$ is commutative. An example of a finite, local, noncommutative ring $ R$ with $ V(R)$ 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

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