Relatively congruence modular quasivarieties of modules

TL;DR

This paper characterizes the quasiequational theory of relatively congruence modular quasivarieties of modules over a ring using ideals and filters, revealing their structure and axiomatizability.

Contribution

It establishes a correspondence between the theory of such quasivarieties and algebraic structures like ideals and filters, providing new insights into their axiomatization.

Findings

The theory is determined by a two-sided ideal and a filter of left ideals.

If R is left Artinian, quasivarieties are axiomatizable by identities and at most one quasiidentity.

For commutative Artinian rings, these quasivarieties are actually varieties.

Abstract

We show that the quasiequational theory of a relatively congruence modular quasivariety of left $R$-modules is determined by a two-sided ideal in $R$ together with a filter of left ideals. The two-sided ideal encodes the identities that hold in the quasivariety, while the filter of left ideals encodes the quasiidentities. The filter of left ideals defines a generalized notion of torsion. It follows from our result that if $R$ is left Artinian, then any relatively congruence modular quasivariety of left $R$-modules is axiomatizable by a set of identities together with at most one proper quasiidentity, and if $R$ is a commutative Artinian ring then any relatively congruence modular quasivariety of left $R$-modules is a variety.