Finite rings in varieties with definable principal congruences

Simons, G. E.

doi:10.1090/S0002-9939-1994-1207541-5

Finite rings in varieties with definable principal congruences
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Proc. Amer. Math. Soc. 121 (1994), 649-655 Request permission

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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