# injective hull

Let $X$ and $Q$ be modules.
We say that $Q$ is an injective hull or injective envelope of $X$
if $Q$ is both an injective module^{} and an essential extension^{} of $X$.

Equivalently, $Q$ is an injective hull of $X$
if $Q$ is injective,
and $X$ is a submodule^{} of $Q$,
and if $g:X\to {Q}^{\prime}$ is a monomorphism^{}
from $X$ to an injective module ${Q}^{\prime}$,
then there exists a monomorphism $h:Q\to {Q}^{\prime}$
such that $h(x)=g(x)$ for all $x\in X$.

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

Every module $X$ has an injective hull, which is unique up to isomorphism^{}. The injective hull of $X$ is sometimes denoted $E(X)$.

Title | injective hull |
---|---|

Canonical name | InjectiveHull |

Date of creation | 2013-03-22 12:10:05 |

Last modified on | 2013-03-22 12:10:05 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 7 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 16D50 |

Synonym | injective envelope |