## Describe M/Tor(M) for a finitely generated module M over a PID

Let $M$ be a finitely generated module over a principal ideal domain $R$. Describe the structure of $M/\mathsf{Tor}(M)$.

We begin with a lemma.

Lemma: Let $R$ be a domain, and let $M$ and $N$ be (left, unital) $R$-modules. Then $\mathsf{Tor}(M \oplus N) = \mathsf{Tor}(M) \oplus \mathsf{Tor}(N)$. Proof: $(\subseteq)$ Let $(m,n) \in \mathsf{Tor}(M \oplus N)$. Then there exists a nonzero element $r \in R$ such that $r(m,n) = (0,0)$. Then $(rm,rn) = (0,0)$, and so $rm = 0$ and $rn = 0$. So $m \in \mathsf{Tor}(M)$ and $n \in \mathsf{Tor}(N)$. $(\supseteq)$ Let $(m,n) \in \mathsf{Tor}(M) \oplus \mathsf{Tor}(N)$. Then there exist nonzero elements $r,s \in R$ such that $rm = 0$ and $sn = 0$. Now $rs$ is nonzero in $R$ (since it is a domain), and we have $rs(m,n) = (s(rm), r(sn)) = 0$. So $(m,n) \in \mathsf{Tor}(M \oplus N)$. $\square$

Now by the Fundamental Theorem of Finitely Generated Modules over a PID (FTFGMPID), if $M$ is such a module, we have $M \cong_R R^t \oplus \bigoplus R/(p_k^{e_k})$ for some primes $p_k \in R$ and natural numbers $e_k$. Now $\mathsf{Tor}(M) = \mathsf{Tor}(R^t) \oplus \mathsf{Tor}(\bigoplus R/(p_k^{e_k}))$ $= \bigoplus R/(p_k^{e_k})$, since $R^t$ is torsion-free (since $R$ is a domain) while… the stuff inside the big sum is all torsion.

Using this previous exercise, we have that $M/\mathsf{Tor}(M) \cong R^t$.