projective cover

Let $X$ and $P$ be modules.We say that $P$ is a projective cover of $X$ if $P$ is a projective moduleand there exists an epimorphism $p\\colon P\\to X$ such that $\\operatorname{ker}p$ is a superfluous submodule of $P$ .

Equivalently, $P$ is an projective cover of $X$ if $P$ is projective,and there is an epimorphism $p\\colon P\\to X$ ,and if $g\\colon P^{\\prime}\\to X$ is an epimorphismfrom a projective module $P^{\\prime}$ to $X$ ,then there exists an epimorphism $h\\colon P^{\\prime}\\to P$ such that $ph=g$ .

\\xymatrix{&P^{\\prime}\\ar[d]^{g}\\ar@{-->}[dl]_{h}\\\\P\\ar[r]_{p}&X\\ar[r]\\ar[d]&0\\\\&0}