Congruences and ideals on Boolean modules: a heterogeneous point of view

Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module M are given and the respective lattices CongM and IdeM are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between CongM and IdeM. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim