# Strictly invariant submodules

## TL;DR

This paper investigates strictly invariant submodules of modules, especially Abelian groups, showing that under certain conditions these are also strongly invariant, invariant under all homomorphisms.

## Contribution

It introduces and analyzes the concept of strictly invariant submodules and establishes conditions under which they are also strongly invariant in Abelian groups.

## Key findings

- Strictly invariant submodules are characterized in various modules.
- In many cases, strictly invariant submodules are also strongly invariant.
- Results apply specifically to Abelian groups and their submodule invariance properties.

## Abstract

If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called \textsl{strictly invariant}. For the case of $% \mathbb{Z}$-modules (i.e. Abelian groups) we prove that in many situations these submodules are invariant with respect to all homomorphisms $% K\rightarrow M$, submodules which were called \textsl{strongly invariant}.