Is $R$ a PID if every submodule of a free $R$-module is free?

Question

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D.

Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is free, is $R$ a PID?

I couldn't find my second question here, only the first one.

Another question regarding this if it's true is how much can we "stretch" the conditions, this is, could be concluded that $R$ is a PID knowing that some submodules of a free $R$-module verify some condition? "Some condition" may sounds vague, but I couldn't think of any that will do.

Answer 1

Yes, it is.

If $a\in R$, $a\ne 0$, then $aR$ is free (as a submodule of the free $R$-module $R$), so $a$ is a non-zero divisor. In order to prove this let $ax$ be an element of an $R$-basis of $aR$. If $ba=0$ for some $b\in R$ then $b(ax)=0$ and therefore $b=0$. (For a characterization of free ideals see here.) This shows that $R$ is necessarily an integral domain.
Now let $I$ be a non-zero ideal of $R$. Since $I$ is free (as a submodule of the free $R$-module $R$) it must have a basis with a single element (as you already know), so $I$ is principal.

(As you may noticed, I've only used the property for the free $R$-module $R$.)

Answer 2

We know that $R$ is a free $R$-module with basis $X=\{1\}$. Now let $I$ be an ideal of $R$.(the submodules of a ring are exactly its ideals.) By hypothesis $I$ has a basis $X$. We claim that $X$ has exactly one element. If $a$ and $b$ are two distinct elements of $X$, then $ab-ba=0$, since $R$ is commutative, a contradiction to the fact that $X$ is linearly independent. Therefore $X$ has only one nonzero element, i.e. $I$ is a principal ideal.