zero ideal

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


and called the zero idealMathworldPlanetmathPlanetmath.

The zero ideal is the identity elementMathworldPlanetmath in the addition of ideals and the absorbing element in the multiplication of ideals (  The quotient ringMathworldPlanetmath R/(0) is trivially isomorphicPlanetmathPlanetmathPlanetmath to R.

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

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