Let $R$ be a PID and $F$ be a free module over $R$. Suppose we have $F=A\oplus B$ for some $R$-modules $A$ and $B$. Then are $A$ and $B$ necessarily free?

If $F$ is finitely generated, then I know that $A$ too is finitely generated. Thus we can write $A$ as a direct sum of a torsion module and a free module. Since there is no torsion in $F$, the torsion part of $A$ must be zero. Thus $A$ must be free and same for $B$.

Is the statement true even when $F$ is not finitely generated?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.