---
tag: 6ANP035KG8
kind: definition
name: Self-injective ring
msc: [16D50]
defines: KQKJENEMSN
uses: [0B1261B5ED, 0NZ8JA25WK, 85H7PRHDPZ]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-Frobenius_ring
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The ring $R$ is *left self-injective* if $R$, viewed as a left [[module|0NZ8JA25WK]] over itself by left multiplication, is an [[injective module|85H7PRHDPZ]].
It is *right self-injective* if $R$ is injective as a right module over itself, and *self-injective* if it is both.

## Remarks

Example: $R = \Z/n\Z$ for $n \ge 1$ is self-injective.
By Baer's criterion it suffices to extend maps $f$ from an ideal $d\Z/n\Z$, with $d \mid n$, to $R$.
The image $a = f(d)$ satisfies $(n/d) a = 0$, so $a = db$ for some $b \in R$, and $x \mapsto bx$ extends $f$.
Non-example: $\Z$ is not self-injective.
The map $2\Z \to \Z$, $2m \mapsto m$, has no extension $g \colon \Z \to \Z$, since $2 g(1) = 1$ has no solution in $\Z$.
Left and right self-injectivity are different conditions in general.
A ring is [[quasi-Frobenius|KP1PP7A6JR]] if and only if it is left Noetherian and left self-injective.