I am interested in the two following statements:
Let $D$ be a principal ideal domain and let $M$ be a $D$-module.
$\bullet$ Let $K\leq M$ be a submodule. $M$ is a torsion module if and only if $K$ and $M/K$ are both torsion modules.
$\bullet$ Let $H,K\leq M$ be two submodules. $H+K$ is a torsion module if and only if $H$ and $K$ are both torsion modules.
I understand that both results are quite related but I don't know how to prove them. Could someone tell me some source where to find these demonstrations or tell me how to do them?
M is a torsion module implies that K and M/K are torsion modules trivially. Suppose that K and M/K are torsion modules. Then for every $x \in M$ there exists a $r \in D$ such that $rx \in K$ (i.e. is zero in M/K). Since $rx \in K$ there exists $s \in D$ such that $srx=0$, therefore (sr)x=0, D is a domain, so $ (sr)$ is not zero, thus x is a torsion element.
H + K is a torsion module, then. for any $k \in K $ $k \in H+ K$ (k +0 = k). If H and K are torsion modules, for any $k \in K $ $h \in H$ there exists $r,s \in D$ such that $sk =0$, $rh=0$, so $rs(h + k) =0$, therefore H+K is a torsion module.