# projective module

A module $P$ is projective
if it satisfies the following equivalent^{} conditions:

(a) Every short exact sequence^{}
of the form $0\to A\to B\to P\to 0$
is split (http://planetmath.org/SplitShortExactSequence);

(b) The functor^{} $\mathrm{Hom}(P,-)$
is exact (http://planetmath.org/ExactFunctor);

(c) If $f:X\to Y$ is an epimorphism^{}
and there exists a homomorphism^{} $g:P\to Y$,
then there exists a homomorphism $h:P\to X$
such that $fh=g$.

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

(d) The module $P$ is a direct summand^{} of a free module^{}.

Title | projective module^{} |
---|---|

Canonical name | ProjectiveModule |

Date of creation | 2013-03-22 12:09:42 |

Last modified on | 2013-03-22 12:09:42 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 7 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16D40 |

Related topic | InvertibleIdealsAreProjective |