zero ideal

The **zero ideal** or **trivial ideal** of a ring $R$ is the two-sided ideal that consists entirely of the zero element. It may be denoted $\{0\}$, $\mathbf{0}$, or simply $0$ (since it is the zero element of the rig of ideals?).

We may generalize to a rig, including the special case of a distributive lattice (in which the zero element is the bottom element), then generalize further to any poset with a bottom element $\bot$, in which the trivial ideal is $\{\bot\}$.

The trivial ideal of $R$ is the intersection of all of the ideals of $R$. (If $R$ is a poset *without* a bottom element, then we may still consider the intersection of all of its ideals, but I'm not sure if this deserves the name.)

Last revised on November 7, 2017 at 19:29:27. See the history of this page for a list of all contributions to it.