prime ideal
Let $R$ be a ring. A twosided proper ideal^{} $\U0001d52d$ of a ring $R$ is called a prime ideal^{} if the following equivalent^{} conditions are met:

1.
If $I$ and $J$ are left ideals^{} and the product of ideals $IJ$ satisfies $IJ\subset \U0001d52d$, then $I\subset \U0001d52d$ or $J\subset \U0001d52d$.

2.
If $I$ and $J$ are right ideals with $IJ\subset \U0001d52d$, then $I\subset \U0001d52d$ or $J\subset \U0001d52d$.

3.
If $I$ and $J$ are twosided ideals with $IJ\subset \U0001d52d$, then $I\subset \U0001d52d$ or $J\subset \U0001d52d$.

4.
If $x$ and $y$ are elements of $R$ with $xRy\subset \U0001d52d$, then $x\in \U0001d52d$ or $y\in \U0001d52d$.
$R/\U0001d52d$ is a prime ring^{} if and only if $\U0001d52d$ is a prime ideal. When $R$ is commutative^{} with identity^{}, a proper ideal $\U0001d52d$ of $R$ is prime if and only if for any $a,b\in R$, if $a\cdot b\in \U0001d52d$ then either $a\in \U0001d52d$ or $b\in \U0001d52d$. One also has in this case that $\U0001d52d\subset R$ is prime if and only if the quotient ring^{} $R/\U0001d52d$ is an integral domain.
Title  prime ideal 

Canonical name  PrimeIdeal 
Date of creation  20130322 11:50:54 
Last modified on  20130322 11:50:54 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  15 
Author  djao (24) 
Entry type  Definition 
Classification  msc 16D99 
Classification  msc 13C99 
Related topic  MaximalIdeal 
Related topic  Ideal 
Related topic  PrimeElement 