---
tag: QCJN05GN64
kind: concept
name: Injective hull
msc: [13C11, 18G05]
sources:
  - url: https://en.wikipedia.org/wiki/Injective_hull
same_as:
  wikidata: Q6034324
  wikipedia: Injective hull
---

## Summary

The injective hull of a module $M$ is the smallest [[injective module|85H7PRHDPZ]] that contains $M$.
Every module has one, and it is unique up to an isomorphism that fixes $M$.
Injective hulls are the building blocks of minimal injective resolutions.
The injective hull of the residue field of a local ring is the dualizing object of [[Matlis duality|2V9BR5XX6A]].

## Explanations

### L1 human

An injective module is one into which maps can always be extended.
The injective hull is the most economical way to enlarge $M$ into such a module.
Every nonzero piece of the hull touches $M$ itself, so nothing superfluous is added.
For example, the injective hull of the integers $\Z$, as an abelian group, is the group of rational numbers $\Q$.