Why an ideal in a ring is a submodule of a free module over that ring I know an ideal in a ring is a module over this ring, but I don't know why it's a submodule of a free module, what's the free module? Thank you.

A fintely generated free $R$-module is isomorphic to $R^n$. Take $n=1$ and you see that $R$ is a free $R$-module. Now it is left to check that an ideal $I\subset R$ defines an $R$-submodule. But this is clear by definition. Is every step and the result clear to you?
