Preenveloping Classes of Acts

TL;DR

This paper explores the concept of preenvelopes in acts over monoids, establishing conditions under which certain classes are preenveloping and providing examples like absolutely pure and weakly injective acts.

Contribution

It characterizes preenveloping classes of acts over monoids as those closed under direct products, extending the analogy with module theory.

Findings

Preenvelopes of acts are defined similarly to modules.

A class of acts is preenveloping if and only if it is closed under direct products.

Examples include absolutely pure, weakly f-injective, and weakly p-injective acts.

Abstract

Preenvelopes of acts over a monoid are defined by analogy with Enochs' definition of preenvelopes of modules. Provided that it is closed for pure subacts, a class of acts is shown to be preenveloping precisely when it is closed under direct products. Examples of such classes include absolutely pure, weakly f-injective and weakly p-injective acts.