Research Papers

On the local finite separability of finitely generated commutative rings  

S. I. Kublanovskii

Creative Production Association "Northern Hearth"

Abstract:   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).

Keywords: residual finiteness, finite separability, commutative rings, profinite topology, subrings closed in profinite topology.

Received: 13.02.2024