Multiplicative lattices and frames (Q1091409)

The author considers multiplicative lattices, that is complete lattices equipped with an associative binary operation which is distributive over arbitrary joins. As is well known, frames are the special case of multiplicative lattices in which the multiplication is idempotent and has the top element of the lattice as a two-sided unit. Recently there has been some interest in studying certain non-commutative multiplicative lattices (christened quantales by C. J. Mulvey) as a non-commutative generalization of frames, but the present paper is concerned mainly with the commutative case. The main result is an extension to multiplicative lattices whose underlying lattice is continuous of a classical result of W. Krull concerning ideal lattices of commutative rings.