nLab 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.)