Fractional elements in multiplicative lattices
Richard G. Burton · Pacific Journal of Mathematics · 1975
An abstract study of the theory of fractional ideals of a commutative ring is begun.In particular, the definition of principal element in a multiplicative lattice L is used to define a lattice of fractional elements, L*, associated with L. As one application of this definition a theory of Dedekind lattices is developed.This construction also allows the development of an abstract theory of integral closure for a Noether lattice.This theory will be presented in a further paper.By a multiplicative lattice we mean a complete lattice L together with a commutative, associative multiplication on L such that (i) a(b U c) = ab U ac and (ii) ab S a Π b for all α, b, c in L. We further assume that L has a greatest element e such that ea = a for all a in L and a least element 0. We denote the meet and join of two elements α, b in L by a U b and a Π b, respectively, and we use ^ to denote the order relation on L. A lattice with a multiplication satisfying condition (i) above is a lattice ordered semi-group.An element m in L is join principal if (a U bm): m = a: m U b for all a, b in L, meet principal if (a Π b: m)m = am Γϊ b for all α,b in L, and principal if it is both join and meet principal.This definition of principal element was given by Dilworth in [1].