Let $A$ be a finitely generated Abelian group, i.e., $$A≃ℤ^{r}⊕C$$ where $C$ is a finite abelian group. Let $B$ be another finitely generated Abelian group, i.e., $$B≃ℤ^{m}⊕C$$ where $m$ and $r$ are unrelated, but we can assume that $m≤r$ or $r≤m$.
My question is: If $r≤m$, can we deduce that $A\leq B$?, i.e., $A$ is a subgroup of $B$.
If $r\leqslant m$; we have an injective group morphism $\Bbb Z^r\hookrightarrow \Bbb Z^m $, in turn we have the identity isomorphism $C\to C$, which we can glue together to get an injective morphism $A\hookrightarrow B$.