##  [Residually Finite Algebra](/index.php/residually-finite-algebra) 

  ##  [Residually Finite Algebra](https://puremath.quantumdictionary.io/residually-finite-algebra-0) 

  

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

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra for which any two distinct elements can be separated by a homomorphism to some finite algebra; equivalently the intersection of all finite-index congruences is the trivial congruence.

 

 

 

 

 





 

 



 ##  [Residually Finite Algebra](https://algebra.quantumdictionary.io/residually-finite-algebra-1) 

  

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

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra is residually finite if for every pair of distinct elements there exists a homomorphism from the algebra to some finite algebra that separates them (their images are distinct); equivalently the algebra embeds into a direct product of finite algebras.