Let $(R,\mathfrak m, k)$ be an Artinian local ring. So for every non-zero finitely generated $R$-module $M$, we have $\mathfrak m\in \mathrm{Ass}(M)$, hence we have an exact sequence $0\to k\to M$, so in particular, $0\to k\to R$.
My question is:
If $0\to A\to B\to C\to 0$ is an exact sequence of finitely generated $R$-modules such that $A,C$ are torsion-less, then is $B$ also torsion-less ?
Here, a finitely generated module $M$ is called torsion-less iff it embeds into a free module of finite rank, or equivalently, if the canonical map $M\to M^{**}$ is injective.