Envelopes of commutative rings

TL;DR

This paper investigates the conditions under which commutative rings have envelopes within specific classes such as fields, semisimple rings, or integral domains, providing full characterizations in these cases.

Contribution

It offers a comprehensive analysis of envelopes of commutative rings for various classes, including new results for Noetherian and non-Noetherian rings.

Findings

Full characterization for fields, semisimple rings, and integral domains.

Results for zero-dimensional Noetherian rings with epimorphic envelopes.

Reduction of the general problem to non-Noetherian rings with monomorphic envelopes.

Abstract

Given a significative class $F$ of commutative rings, we study the precise conditions under which a commutative ring $R$ has an $F$-envelope. A full answer is obtained when $F$ is the class of fields, semisimple commutative rings or integral domains. When $F$ is the class of Noetherian rings, we give a full answer when the Krull dimension of $R$ is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.