Comparing $\mathrm{Add}(M)$ with $\mathrm{Prod}(M)$

TL;DR

This paper characterizes when classes of modules formed by additive and product constructions coincide or include each other in certain categories, with applications to covering and enveloping classes.

Contribution

It provides new characterizations of the inclusion relations between Add(M) and Prod(M) in specific categories, advancing understanding of their structural properties.

Findings

Characterizations of Add(M) and Prod(M) inclusions in categories

Conditions for classes to be (pre)covering or (pre)enveloping

Applications to module theory and triangulated categories

Abstract

We present characterizations for the inclusions $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$ and $\mathrm{Prod}(M)\subseteq \mathrm{Add}(M)$ in locally finitely presented categories and in compactly generated triangulated categories. As applications, we describe the situations when the classes of the form $\mathrm{Prod}(M)$ and $\mathrm{Add}(M)$ are (pre)covering, respectively (pre)enveloping.