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
Canonical name	ProjectiveCover
Date of creation	2013-03-22 12:10:08
Last modified on	2013-03-22 12:10:08
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	6
Author	antizeus (11)
Entry type	Definition
Classification	msc 16D40