# injective module

A module $Q$ is an injective module^{}
if it satisfies the following equivalent^{} conditions:

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

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

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

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

Title | injective module |
---|---|

Canonical name | InjectiveModule |

Date of creation | 2013-03-22 12:02:26 |

Last modified on | 2013-03-22 12:02:26 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 8 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16D50 |