Lattice theory for certain classes of rings

Abstract

We observe that every Boolean ring is aP1-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP2-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P1-rings andP2-rings are generalisations of a Boolean ring in terms of its lattice structure. Then, it is only natural that these two classes of rings have such close affiliations with the lattice structure of a Boolean ring.