ideal
Let $R$ be a ring. A left ideal^{} (resp., right ideal) $I$ of $R$ is a nonempty subset $I\subset R$ such that:

•
$ab\in I$ for all $a,b\in I$

•
$r\cdot a\in I$ (resp. $a\cdot r\in I$) for all $a\in I$ and $r\in R$
A twosided ideal is a left ideal $I$ which is also a right ideal. If $R$ is a commutative ring, then these three notions of ideal are equivalent^{}. Usually, the word “ideal” by itself means twosided ideal.
The name “ideal” comes from the study of number theory^{}. When the failure of unique factorization^{} in number fields was first noticed, one of the solutions was to work with socalled “ideal numbers” in which unique factorization did hold. These “ideal numbers” were in fact ideals, and in Dedekind domains^{}, unique factorization of ideals does indeed hold. The term “ideal number” is no longer used; the term “ideal” has replaced and generalized it.
Title  ideal 
Canonical name  Ideal 
Date of creation  20130322 11:49:27 
Last modified on  20130322 11:49:27 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  18 
Author  djao (24) 
Entry type  Definition 
Classification  msc 14K99 
Classification  msc 16D25 
Classification  msc 11N80 
Classification  msc 13A15 
Classification  msc 54C05 
Classification  msc 54C08 
Classification  msc 54J05 
Classification  msc 54D99 
Classification  msc 54E05 
Classification  msc 54E15 
Classification  msc 54E17 
Related topic  Subring 
Related topic  PrimeIdeal 
Defines  left ideal 
Defines  right ideal 
Defines  2sided ideal 
Defines  twosided ideal 