Geodesic
Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents
Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 45-47

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It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors. 
Cornish, William H. Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents. Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 45-47. doi: 10.4153/CMB-1973-010-4
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