##  [Nilradical](/index.php/nilradical) 

  ##  [Nilradical](https://puremath.quantumdictionary.io/nilradical-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

In a commutative ring R, the nilradical Nil(R) is the ideal consisting of all nilpotent elements of R; equivalently it is the intersection of all prime ideals of R. It captures the nonreduced part of the ring.

 

 

 

 

 





 

 



 ##  [Nilradical](https://algebra.quantumdictionary.io/nilradical-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

The ideal of a ring R consisting of all nilpotent elements; equivalently the set {a in R : a^n = 0 for some n>0} which is an ideal and equals the intersection of all prime ideals of R.