Varieties of rings with definable principal congruences

Simons, G. E.

doi:10.1090/S0002-9939-1983-0684626-6

Varieties of rings with definable principal congruences
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by G. E. Simons PDF

Proc. Amer. Math. Soc. 87 (1983), 397-402 Request permission

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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