On the local finite separability of finitely generated commutative rings

Necessary and sufficient conditions are exhibited for the local finite separability of finitely generated commutative rings by reducing their description to the case of rings of prime characteristic without zero divisors. As a corollary, it is shown that, in contrast to the situation for groups, the class of these rings is closed under homomorphic images and finite direct products. It is also proved that a finitely generated commutative ring is locally finitely separable if and only if so is each of its two-generated subrings. It is shown 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).