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# 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. 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.

Related:

MinimalPrimeIdeal, PrimeRing, ZeroModule

Type of Math Object:

Definition

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## Mathematics Subject Classification

14K99*no label found*16D25

*no label found*11N80

*no label found*13A15

*no label found*

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