 ##  [Primary Decomposition](/primary-decomposition) 

  ##  [Primary Decomposition](https://mathlogic.quantumdictionary.io/primary-decomposition-0) 

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

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

The expression of an ideal (or a submodule) in a Noetherian ring as a finite intersection of primary ideals (or primary submodules), each of which has a radical that is a prime ideal; associated primes appear as radicals of the primary components.

 
 ##  [Primary Decomposition](https://puremath.quantumdictionary.io/primary-decomposition-1) 

  
 [![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 &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

The expression of an ideal I in a Noetherian ring (commonly a polynomial ring) as a finite intersection I = Q1 ∩ … ∩ Qr of primary ideals Qi, where each Qi has a radical Pi that is prime; this decomposition isolates the primary components corresponding to the distinct prime-associated geometric components of V(I).

 
 ##  [Primary Decomposition](https://algebra.quantumdictionary.io/primary-decomposition-2) 

  
 [![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 expression of a suitably finitary substructure (typically an ideal in a Noetherian ring) as an intersection of primary substructures, where each primary component has an associated prime controlling its nilpotent behavior.