A question from the proof of: If M is a free f.g R-module, and N is a submodule of M, then N is free, where R is a PID

Question

I'm reading the proof from Dummit and Foote third edition, page 460-461,theorem 4.

I am confused about one thing: On page 461, there are:

• (a) $M=Ry_1\oplus \ker v$
• (b)$N=Ra_1y_1\oplus (\ker v\cap N)$.

But as M is free(and N will be proved as free), so $Ry_1$ should be isomorphic to R and so does $Ra_1y_1$. So it means the cyclic modules $Ry_1$ and $Ra_1y_1$ are free, but this is not obvious to me(to most cyclic modules they are isomorphic to $R/I$, and I don't have a easy way to see they are free.

Do you have some easy way to see they are free? Thank you!

Answer 1

I don't have the book by Dummit and Foote, but the maps $$ R\to Ry_1,\quad r\mapsto ry_1\\ R\to Ra_1y_1,\quad r\mapsto ra_1y_1 $$ are surjective homomorphisms and they're also injective because $M$ is torsion-free.

For the second one, note that $ra_1=0$ implies $r=0$, if $a_1\ne0$. If $a_1=0$, then $Ra_1y_1=\{0\}$ is free as well.