Is this a projective resolution? Let $k$ be a field, $R=k[x]$. Is this a projective resolution of $k$ over $R$?
$$0\to k[x]\to k[x]\to k\to 0$$
where the left map is $x\mapsto x-1$ and the right map is $x\mapsto 1$ ?
If not, what is a projective resolution in this case?

Let $f:k[x]\rightarrow k[x]$ and $g:k[x]\rightarrow k$, $g(f(1))=1\neq 0$, thus it is not a resolution.
