Ideals, Natural Classes, and Functors

Abstract

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) ⊕ 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) ≅ 𝒩 f (R) is recovered.