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

Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module $mathcal {M}$ are given and the respective lattices $mathrm{Cong}mathcal {M}$ and $mathrm{Ide}mathcal {M}$ 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 $mathrm{Cong}mathcal {M}$ and $mathrm{Ide}mathcal {M}$. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined.