Let $E=ℤ^{r}⊕T$ be a finitely generated abelian group. $T$ is a finite abelian group of rank $s$.
My question is about the rank of $E$. I thought that it is equal to $r+s$ but I am not sure about this result.
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0 You are mixing up the terms rank (see here) and order (see here). For any two abelian groups $A$ and $B$, it is true that $$\mathrm{rank}(A\oplus B)=\mathrm{rank}(A)+\mathrm{rank}(B),$$ but the rank of any finite abelian group is $0$, so the rank of $E$ in your question is still $r$.
