minimal projective presentation

Let $R$ be a ring and $M$ a (right) module over $R$. A short exact sequence of modules

is called a minimal projective presentation of $M$ if both $p_0:P_0\to M$ and $p_1:P_1\to \ker p_0$ are projective covers.

Minimal projective presenetations are unique in the following sense: if

are both minimal projective presentations of $M$, then this diagram can be completed to the following commutative one:

were both $a,b$ are isomorphisms.

It can be shown, that if $R$ is a finite-dimensional algebra over a field $k$, then every finitely generated $R$-module $M$ admits minimal projective presentation (indeed, $R$ is semiperfect (http://planetmath.org/PerfectAndSemiperfectRings) in this case).