The monograph addresses research mathematicians and graduate students interested in the module and representation theory of arbitary rings. It is primarily concerned with generalizations of injectivity and projectivity, and simultaneously with modules displaying good direct decomposition properties. Specifically, we study two classes of modules, named continuous and discrete. Both exhibit, in a dual sense, a generous supply of direct summands. The first class contains all injective modules, while the second one contains those projective modules which have a “good” direct sum decomposition.

Continuous, as the term is used here, is not related to continuity in the sense of topology and analysis. It is rather derived from the notion of a continuum. This usage originated with von Neumann's continuous geometries. These are analogues of projective geometries, except that they have no points, but instead a dimension function whose range is a continuum of real numbers. Just as most projective geometries can be coordinatized by simple artinian rings, most continuous geometries are coordinatized by non-noetherian continuous regular rings.