Consider the free abelian group $\mathbb Z^n$, with elements considered as row vectors. For every $A\in M_{r\times n}(\mathbb Z)$ , let $K_A$ be the subgroup of $\mathbb Z^n$ generated by the row vectors of $A$ . Now let $A\in M_{r\times n}(\mathbb Z)$ and $B:=PAQ$ where $P\in GL_{r\times r}(\mathbb Z), Q \in GL_{n\times n}(\mathbb Z)$. Then how to prove that $\mathbb Z^n/K_A $ and $\mathbb Z^n/K_B$ are isomorphic as groups ?
I think the map $f: \mathbb Z^n/K_A \to \mathbb Z^n/K_B$ defined as $f(\bar a +K_A)=\bar a Q+K_B, \forall \bar a \in \mathbb Z^n$ is an isomorphism, but I am not sure (I can't even show whether this map is well-defined or not).
Please help.