Conclusion
We observe that every Boolean ring is aP 1-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP 2-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P 1-rings andP 2-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.
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Subrahmanyam, N.V. Lattice theory for certain classes of rings. Math. Ann. 139, 275–286 (1960). https://doi.org/10.1007/BF01352263
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DOI: https://doi.org/10.1007/BF01352263