On the local finite separability of finitely generated commutative rings

  We exhibit necessary and sufficient conditions for the local finite separability of finitely generated commutative rings, reducing their description to the case of rings of prime characteristic without zero divisors. As a corollary, we show that, in contrast to the situation for groups, the class of these rings is closed under homomorphic images and finite direct products. We also prove that a finitely generated commutative ring is locally finitely separable if and only is so is each of its two-generated subrings. We show that two-generated commutative rings of nonzero characteristic whose generators are subject to a nontrivial homogeneous defining relation are locally finitely separable (consequently, such rings have a decidable membership problem for finitely generated subrings).