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.