How does a free resolution of a module fail to be finite?

Question

Let $M$ be any $R$-module.

We can construct a free resolution as follows:

Let $M_1$ be the free module with basis in one to one correspondence with generators of $M$. Then the map $\phi:M \to M_1$ is necessarily surjective. Let $M_2 = \operatorname{ker}(\phi)$.

I don't see how the sequence : $0 \to M_2 \to M_1 \to M \to 0$ fails to be a finite resolution of $M$?

I am claiming that the map $M_2 \to M_1$ is the inclusion map and thus has kernel $0$.

Answer 1

If $I$ is a nonprincipal ideal of a ring, then $0\to I\to R\to R/I\to 0$ is a counterexample.

If $I$ were free and generated by more than one basis elements $u,v$, then $uv -vu=0$ contradicts the basis property.

In fact, the property of having all right ideals free in a commutative ring characterizes PID's. That is why it works for Abelian groups (since they are just $\mathbb Z$ modules.)