##  [Polynomial Identity (PI) Condition](/polynomial-identity-pi-condition) 

  ##  [Polynomial Identity (PI) Condition](https://algebra.quantumdictionary.io/polynomial-identity-pi-condition-0) 

  

 [![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

The property that an associative algebra A satisfies a nonzero polynomial identity: there exists a nonzero polynomial f in noncommuting variables such that f(a1,...,an)=0 for all substitutions of elements ai in A.