##  [Hilbert Basis Theorem](/hilbert-basis-theorem) 

  ##  [Hilbert Basis Theorem](https://puremath.quantumdictionary.io/hilbert-basis-theorem-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

A theorem stating that if R is a Noetherian ring (commutative with unity, typically), then the polynomial ring R[x] is also Noetherian; by induction, R[x1,...,xn] is Noetherian for every finite n. Equivalently, ideals in such polynomial rings are finitely generated.

 

 

 

 

 





 

 



 ##  [Hilbert Basis Theorem](https://algebra.quantumdictionary.io/hilbert-basis-theorem-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

A theorem asserting that if R is a Noetherian commutative ring then the polynomial ring R[x1,…,xn] in finitely many indeterminates over R is also Noetherian; equivalently, every ideal of R[x1,…,xn] is finitely generated when R satisfies the ascending chain condition on ideals.