IDEALS, NATURAL CLASSES, AND FUNCTORS

For any ring R, the set N(R) of all natural classes of R-modules is a complete Boolean lattice, which is a direct sum of two convex and complete Boolean sublattices N(R) = N-t (R) circle plus N-f(R), where the last summand is the set of all nonsingular natural classes. The ring R contains a unique lattice of ideals J(R) which is lattice isomorphic to N-f(R). The present note develops the analogue of all of the above for an arbitrary R-module M, so that in the special case when M-R = R-R, the known lattice isomorphism J(R) congruent to N-f(R) is recovered.