I want to prove this example from Rotman's which I have not found proved in literature yet and Im curious about. Let $_{R}M$ be a left module over a left herditary ring $R$, then $p.d(_{R}M) \leq 1$,i.e, projective dimension of $_{R} M$ is less than or equal to one.
So here we go writing all we need (I guess), let $R$ be a ring, we say that $R$ is a left hereditary ring if every left ideal $I$ of $R$ is projective as a left module over $R$. According to Rotman´s Homological Algebra we have that the following statements are equivalent:
(i) $R$ is left hereditary ring,
(ii) Submodules of projective left $R$ modules are projective,
(iii) Quotients of injective left $R$ modules are injective.
To prove $p.d(_{R}M)\leq 1$ Rotman suggest using (ii) of the last list of equivalences. So proving this means that for every projective resolution
$$...\rightarrow P_{2} \rightarrow P_{1} \rightarrow P_{0} \rightarrow _{R} M \rightarrow 0 ,$$
$P_{i}=0$ for every $i \geq 1$. But first of all, I need to show I always have $P_{0}$ projective. But I dont have an idea about how to construct this $P_{0}$ and the respective morphism. Any help proving this example will be useful.