zero idealZeroIdeal2009-01-18 08:22:042009-01-18 12:27:02Definitionideal% this is the default PlanetMath preamble. as your knowledge
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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 {\em zero ideal}.\\
The zero ideal is the identity element in the addition of ideals and the absorbing element in the \PMlinkname{multiplication of ideals}{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.