Example 1

Suppose that the ring $A$ is principal; let $M$ be an $A$-module and $(x_i)_{i\in I}$ a generating family of $M$. Denote by $L_0$ the free module $A^{(I)}$, by $(e_i)$ its canonical basis, and define $p:L_0\to M$ by $p(e_i)=x_i$. The morphism $p$ is surjective and its kernel $L_1$ is a free $A$-module by VII, § 3, Corollary 2 to Theorem 1; hence the exact sequence

is a free resolution of $M$ of length 1. If $I$ is finite, $L_0$ and $L_1$ are of finite type.