The generalisation of some invariants of modules to groups with operators

TL;DR

This paper extends classical module invariants to groups with operators, analyzing their behavior in semisimple contexts and introducing morphisms that preserve these invariants for better understanding of semisimplicity.

Contribution

It provides a straightforward generalization of module invariants to groups with operators and introduces morphisms that preserve these invariants, aiding the study of semisimplicity.

Findings

Invariants behave well under restricted direct sums.

Morphisms preserving invariants are suitable for studying semisimplicity.

Generalization applies to semisimple groups with operators.

Abstract

The article $-$ part of a larger thesis which aims to give a detailed description of the generalisation to the category of groups with operators of the classical theory of semisimplicity for modules $-$ presents a straightforward generalisation to groups with operators of a number of invariants well-known in the theory of modules, having a special bearing on phenomena of semisimplicity. The behaviour of these generalised invariants in relation to the fundamental construction of restricted direct sums and in particular in the context of semisimple groups with operators is in particular examined. The article also introduces a particular type of morphisms between groups with operators, morphisms which naturally preserve the generalised invariants in question and thus show themselves to be an adequate notion for the study of semisimplicity in the more general frame considered.