In general we define the projective cover a module $M$ over an arbitrary ring $R$ as a surjective $R$-map $f: P \rightarrow M$ such that $\operatorname{ker}(f)$ is superfluous.
I read (if I recall correctly in Lambek's book) that $f$ is a projective cover if and only if $\operatorname{ker}(f) \subset \operatorname{rad}(P)$. However I don't see why, is this always true or do we need additional assumptions on $P$ or $M$?