Torsion and module over a ring.

Question

I need to solve the following problem:

Show that a ring $R$ is a field iff every $R$-module is a torsion-free module.

The "only if" part is quite easy because if $R$ is a field then every $R$-module is a vector space then is torsion-free.

I'd like some advice to prove the other implication.

Answer 1

If not, then $R$ has a non-trivial ideal. Then $R/I$ has torsion as an $R$-module.