A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $A.$ Consider the category of such couples $(A,B),$ where morphism $f:(A,B)\to (A',B')$ is a homomorphism $f:A\to A'$ such that $f(B)\subseteq B'.$ A couple $(A,B)$ is called *cyclic* if $A$ and $B$ are cyclic groups.

**Question 1:** Is it true that any couple of finitely generated abelian groups $(A,B)$ is isomorphic to a direct sum of cyclic couples?

If $A$ is free, it follows from the Smith normal form theorem. If there was a version of the Smith normal form theorem for arbitrary homomorphisms of finitely generated abelian groups, then, I believe, this result would follow.

**Question 2:** Is there a version of the Smith normal form theorem for arbitrary homomorphisms of finitely generated abelian groups?