  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}.
