##  [Associative Algebra](/associative-algebra) 

  ##  [Associative Algebra](https://puremath.quantumdictionary.io/associative-algebra-0) 

  

 [![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 & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra A over a commutative ring (often a field) that is simultaneously a module over that ring and a ring whose multiplication is associative and bilinear with respect to the scalar action; unital or nonunital variants occur.

 

 

 

 

 





 

 



 ##  [Associative Algebra](https://algebra.quantumdictionary.io/associative-algebra-1) 

  

 [![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 vector space or module over a commutative ring or field equipped with a bilinear multiplication map that is associative: (ab)c = a(bc) for all elements a,b,c. The algebra structure pairs the linear structure with a single associative binary product.