On the finite separability of finitely generated commutative rings

TL;DR

This paper characterizes when finitely generated commutative rings are finitely separable, showing they are finite extensions of specific torsion ideals and describing their structure in detail.

Contribution

It provides necessary and sufficient conditions for finite separability of finitely generated commutative rings, including structural descriptions involving torsion ideals and prime characteristic extensions.

Findings

Every finitely generated commutative ring is a finite extension of its torsion ideal.

The torsion ideal is a subdirect product of a finite ring and zero divisor-free rings.

These rings are extensions of their infinite monogenic subrings.

Abstract

We find necessary and sufficient conditions for the finite separability of finitely generated commutative rings. Namely, we prove that every such ring is a finite extension of its torsion ideal $I_k$ where $k$ is square-free, and $I_k$ is a subdirect product of a finite ring and finitely many zero divisor-free rings of prime characteristic each of which is an entire extension of any of its infinite monogenic subrings.