projective cover

Let $X$ and $P$ be modules. We say that $P$ is a projective cover of $X$ if $P$ is a projective module and 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 epimorphism from a projective module $P^{\prime}$ to $X$, then there exists an epimorphism $h\colon P^{\prime}\to P$ such that $ph=g$.

 $\xymatrix{&P^{\prime}\ar[d]^{g}\ar@{-->}[dl]_{h}\\ P\ar[r]_{p}&X\ar[r]\ar[d]&0\\ &0}$
Title projective cover ProjectiveCover 2013-03-22 12:10:08 2013-03-22 12:10:08 antizeus (11) antizeus (11) 6 antizeus (11) Definition msc 16D40