# 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 .

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