##  [Prime Ideal](/index.php/prime-ideal) 

  ##  [Prime Ideal](https://mathlogic.quantumdictionary.io/prime-ideal-0) 

  

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

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A proper ideal P of a commutative ring R such that the quotient ring R/P is an integral domain; equivalently, P is proper and whenever a product ab lies in P, at least one of a or b lies in P (the complement R\P is multiplicatively closed).

 

 

 

 

 





 

 



 ##  [Prime Ideal](https://algebra.quantumdictionary.io/prime-ideal-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

A proper ideal P of a commutative ring R such that whenever a·b ∈ P then either a ∈ P or b ∈ P; equivalently the quotient R/P is an integral domain.