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I need to solve the following problem:

Show that a ring $R$ is a field iff every $R$-module is 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 solve the other implication.

Thanks a lot!

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

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