---
tag: 8MS51FASMX
kind: concept
name: Essential submodule
msc: [16D10]
sources:
  - url: https://en.wikipedia.org/wiki/Essential_extension
---

## Summary

An essential submodule of a [[module|0NZ8JA25WK]] $M$ is a [[submodule|05WDSN597C]] that meets every nonzero submodule of $M$ in a nonzero element.
One then says that $M$ is an essential extension of it.
Essential submodules are "large" in a lattice sense, even when they are small in size.
Essential extensions are the basis of the theory of injective hulls, uniform dimension and Goldie's theorem.

## Explanations

### L1 human

A submodule is essential when it cannot be avoided: every nonzero piece of the big module overlaps it.
The even integers $2\Z$ are essential in $\Z$, since any nonzero subgroup $a\Z$ contains the nonzero even number $2a$.
In $\Z \oplus \Z$, the first axis $\Z \oplus 0$ is not essential, since the second axis meets it only in zero.
The rational numbers $\Q$ are an essential extension of $\Z$, and $\Q$ is the injective hull of $\Z$.