##  [Commutative Ring](/commutative-ring) 

  ##  [Commutative Ring](https://mathlogic.quantumdictionary.io/commutative-ring-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 & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A ring whose multiplication is commutative: for all a, b in R, a·b = b·a. It retains the additive abelian group structure and distributivity of a ring while adding multiplicative commutativity.

 

 

 

 

 





 

 



 ##  [Commutative Ring](https://puremath.quantumdictionary.io/commutative-ring-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 & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A ring (an additive abelian group equipped with a multiplication) in which the multiplication operation is commutative; many authors also assume the existence of a multiplicative identity (1 ≠ 0).

 

 

 

 

 





 

 



 ##  [Commutative Ring](https://algebra.quantumdictionary.io/commutative-ring-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 ring in which the multiplication operation is commutative, i.e., ab=ba for all elements a and b; typically considered with the same additive abelian group and distributive properties as rings, and often studied with or without a multiplicative identity.