Varieties of rings with definable principal congruences
G. E. Simons · Proceedings of the American Mathematical Society · 1983
A variety V \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 V \mathcal {V} . The key result is that for any ring R R , V ( M n ( R ) ) V({M_n}(R)) does not have DPC if n ⩾ 2 n \geqslant 2 . This allows us to show that if V ( R ) V(R) has DPC, then R R is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring R R , V ( R ) V(R) has DPC if and only if R R is commutative. An example of a finite, local, noncommutative ring R R with V ( R ) V(R) having DPC is given.