 ##  [Schur's Lemma](/schurs-lemma) 

  ##  [Schur's Lemma](https://mathlogic.quantumdictionary.io/schurs-lemma-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A result in representation theory and module theory: any homomorphism between simple modules is either zero or an isomorphism; equivalently, the endomorphism ring of a simple module is a division algebra, and over algebraically closed fields this endomorphism algebra is just the field (scalars).

 
 ##  [Schur's Lemma](https://puremath.quantumdictionary.io/schurs-lemma-1) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

A statement about morphisms between simple (irreducible) objects in module or representation categories: any homomorphism between two simple modules is either zero or an isomorphism; the endomorphism ring of a simple module is a division ring (a skew field).

 
 ##  [Schur's Lemma](https://algebra.quantumdictionary.io/schurs-lemma-2) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

A statement about homomorphisms of simple (irreducible) modules or representations: any nonzero homomorphism between simple modules is an isomorphism; consequently, the endomorphism ring of a simple module is a division ring (and over an algebraically closed field it is just the field of scalars).