 ##  [Ideal](/ideal) 

  ##  [Ideal](https://mathlogic.quantumdictionary.io/ideal-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

A subset I of a ring R that is an additive subgroup and is closed under multiplication by arbitrary elements of R (for all r in R and x in I, rx and xr lie in I); in commutative rings this means r x ∈ I for every r ∈ R and x ∈ I.

 
 ##  [Ideal](https://puremath.quantumdictionary.io/ideal-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

A subset I of a ring R that is an additive subgroup of R and is closed under multiplication by arbitrary elements of R (r·x and x·r lie in I for r in R and x in I). Ideals encode divisibility, congruence, and factorization structure of the ambient ring and are the kernels of ring homomorphisms.

 
 ##  [Ideal](https://algebra.quantumdictionary.io/ideal-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

A subset I of a ring R that is closed under addition and under multiplication by arbitrary elements of R (r·i and i·r lie in I); in commutative rings this means r·i ∈ I for all r ∈ R, i ∈ I.