Abstract
The ring-theoretical concept of semiprime ideal is appropriately defined for lattices. We prove that an ideal I of a lattice L is semiprime iff I is the kernel of some homomorphisms of L onto a distributive lattice with zero. As a corollary we generalize the Prime Separation Theorem to arbitrary lattices. The theory of semiprime ideals is developed here without assuming the axiom of choice. The radical of an ideal is defined as its semiprime closure and we show that the Ultrafilter Principle is equivalent to the statement that every semiprime ideal is representable as an intersection of prime ideals.
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