---
tag: N67R59ZVC4
kind: statement
role: lemma
name: Schur's lemma in abelian categories
msc: ["18E10"]
uses: [Y9CZ3MTNF1, FS1FKAE89H, S417YNEJ6M, Y2NP1JFR86, RVXDSY04KG, YF9F9EG6M3, 0EA5BTA0CX]
about: [Y9CZ3MTNF1]
sources:
  - url: https://en.wikipedia.org/wiki/Schur%27s_lemma
---

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].
An object $S$ of $\mathcal{A}$ is *simple* if $S$ is not a zero object and its only [[subobjects|FS1FKAE89H]] are $0$ and $S$.
Let $S$ and $T$ be simple objects and let $f \colon S \to T$ be a morphism.
Then $f$ is either the [[zero morphism|YF9F9EG6M3]] or an isomorphism.
In particular, the endomorphism ring $\End(S)$ is a [[division ring|0EA5BTA0CX]].

## Remarks

The [[kernel|S417YNEJ6M]] of $f$ is a subobject of $S$, so it is $0$ or $S$.
The image of $f$ is a subobject of $T$, so it is $0$ or $T$.
If $f \ne 0$, then the kernel is not $S$ and the image is not $0$.
So $f$ is a [[monomorphism|Y2NP1JFR86]] and an [[epimorphism|RVXDSY04KG]], and in an abelian category such a morphism is an isomorphism.
For $\End(S)$, every nonzero element is invertible, and $1 \ne 0$ because $S$ is not zero.
In the category of modules over a ring, the simple objects are the simple modules, and one gets the classical Schur's lemma.
The simplicity hypothesis is needed: multiplication by $2$ on the $\Z$-module $\Z$ is nonzero and not an isomorphism, and $\Z$ is not simple since $2\Z$ is a proper nonzero submodule.

## Explanations

### L1 human

A simple object is one that cannot be cut into smaller nonzero pieces.
Between two such objects, a map either kills everything or matches them up perfectly.
There is nothing in between, because the kernel and the image of the map must each be all or nothing.

### L2 human

The lemma is the first step in studying objects of finite length, which are built from simple objects by extensions as in the Jordan–Hölder theorem.
It shows that $\Hom(S, T) = 0$ for non-isomorphic simple objects and that $\End(S)$ is a division ring.
In representation theory over an algebraically closed field $k$, a further argument with eigenvalues gives $\End(S) = k$ for finite-dimensional simple representations; that sharper form needs the extra hypotheses.