Let $R$ be a field. What are the $R$-submodules of $R \times R \ ?$

Question

We know that $R$-submodule of $R$ are left ideals of $R.$

Is it also true that $R$-submodule of $R \times R$ are left ideals of $R \times R\ ?$

Please advise on the correct approach to this problem.

Thank you.

Answer 1

If $R$ is a field, $R$-module means $R$-vector space.

So you're asking what are the linear spaces in a vector space of dimension $2$.