nLab
zero ideal

The zero ideal or trivial ideal of a ring RR is the two-sided ideal that consists entirely of the zero element. It may be denoted {0}\{0\}, 0\mathbf{0}, or simply 00 (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 RR is the intersection of all of the ideals of RR. (If RR 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.)

Revised on November 7, 2017 19:29:27 by Toby Bartels (64.89.52.15)