prime ideal

Let $R$ be a ring. A two-sided proper ideal $\mathfrak{p}$ 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 $I J$ satisfies $IJ\subset\mathfrak{p}$ , then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$ .
2.

If $I$ and $J$ are right ideals with $IJ\subset\mathfrak{p}$ , then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$ .
3.

If $I$ and $J$ are two-sided ideals with $IJ\subset\mathfrak{p}$ , then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$ .
4.

If $x$ and $y$ are elements of $R$ with $xRy\subset\mathfrak{p}$ , then $x\in\mathfrak{p}$ or $y\in\mathfrak{p}$ .

$R/\mathfrak{p}$ is a prime ring if and only if $\mathfrak{p}$ is a prime ideal. When $R$ is commutative with identity, a proper ideal $\mathfrak{p}$ of $R$ is prime if and only if for any $a,b\in R$ , if $a\cdot b\in\mathfrak{p}$ then either $a\in\mathfrak{p}$ or $b\in\mathfrak{p}$ . One also has in this case that $\mathfrak{p}\subset R$ is prime if and only if the quotient ring $R/\mathfrak{p}$ is an integral domain.

Title	prime ideal
Canonical name	PrimeIdeal
Date of creation	2013-03-22 11:50:54
Last modified on	2013-03-22 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