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.

Key Words:

• Boolean lattice
• Complement closure of a submodule
• Complement submodule
• Natural class
• Rational extension of modules
• Second singular submodule

2000 Mathematics Subject Classification:

• Primary 16D40
• Secondary 16P50, 16D99

Notes

Communicated by T. Albu.