Unit and kernel systems in algebraic frames

One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of d-elements. However, unless the frame is already compact there is no largest such quotient. With the additional assumption of disjointification on the frame, one then studies the maximal ideal spaces of these quotients and the relationship to covers of compact spaces. Several applications are considered, with considerable attention to the frame quotients defined by extension of ideals of a commutative ring A to a ring extension; this type of frame quotient is considered both with and without an underlying lattice structure on the rings. Received January 8, 2005; accepted in final form August 28, 2005.