# 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^{} $p:P\to X$
such that $\mathrm{ker}p$ is a superfluous submodule of $P$.

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

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

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 |