zero ideal

The subset $\{0\}$ of a ring $R$ is the least two-sided ideal of $R$.  As a principal ideal, it is often denoted by

$$(0)$$

and called the zero ideal.

The zero ideal is the identity element in the addition of ideals and the absorbing element in the multiplication of ideals (http://planetmath.org/ProductOfIdeals).  The quotient ring $R/(0)$ is trivially isomorphic to $R$.

By the entry quotient ring modulo prime ideal, (0) is a prime ideal if and only if $R$ in an integral domain.

Title	zero ideal
Canonical name	ZeroIdeal1
Date of creation	2013-03-22 18:44:40
Last modified on	2013-03-22 18:44:40
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Definition
Classification	msc 14K99
Classification	msc 16D25
Classification	msc 11N80
Classification	msc 13A15
Related topic	MinimalPrimeIdeal
Related topic	PrimeRing
Related topic	ZeroModule