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An algebra M = ⟨M ; ∨, •, 0, 1⟩ such that: (1) ⟨M ; ∨, 0, 1⟩ is a join semilattice with least element 0 and greatest element 1, (2) ⟨M ; •, 0, 1⟩ is a commutative monoid with identity 1 and zero 0, (3) the operation • distributes over ∨: x • (y ∨ z) = (x • y) ∨ (x • z) for all x, y, z ∈ M , is called a multiplicative semilattice.

If, in the multiplicative semilattice M , multiplication is idempotent (x • x = x), then M is a bounded distributive lattice. If R = ⟨R; +, •, 0, 1⟩ is a unitary commutative ring, then Id R = ⟨Id R; ∨, •, 0, 1⟩ is a multiplicative semilattice, where Id R is the set of ring ideals of R, I ∨ J is the join of the ideals I and J , I • J is the product of the ideals I and J , 0 = {0}, and 1 = R.

Since multiplicative semilattices and bounded distributive lattices are each defined by a set of identities, every multiplicative semilattice M has a maximal distributive lattice quotient, M /Δ. Let Sm R denote the set of semiprime ideals of R, ordered by set inclusion. Under this order, Sm R is a bounded distributive lattice.

Theorem 0.1. M /Δ is isomorphic to Sm R.

I will prove this theorem and discuss some of its implications.