##  [Nilpotent Element](/nilpotent-element) 

  ##  [Nilpotent Element](https://puremath.quantumdictionary.io/nilpotent-element-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 element a of a ring (or algebra) is nilpotent if a^n = 0 for some positive integer n. The least such n is called the index (or exponent) of nilpotence of a.

 

 

 

 

 





 

 



 ##  [Nilpotent Element](https://algebra.quantumdictionary.io/nilpotent-element-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

An element a of a ring or algebra R such that a^n = 0 for some positive integer n; equivalently, an element whose sufficiently high power vanishes and which signals nonreduced or infinitesimal directions in the algebraic structure.