Tertiary decompositions in lattice modules (Q1080874)

The theory of associated prime ideals of an R-module, and of tertiary decompositions, generalizes to L-modules, where L is a complete modular lattice and an L-module M is a complete modular lattice together with an appropriate module action \(p: L\times M\to M\). Given appropriate chain conditions on L and M, the theory of associated prime ideals, existence and uniqueness properties for tertiary decompositions, and a form of the Krull intersection theorem all hold in generalized form. If more stringent conditions apply, the theory reduces to a generalized theory of primary decompositions and a second uniqueness theorem holds. The theory can be applied to congruence lattices of algebras in congruence-modular varieties of algebras, using the generalized commutator operation. An important special case is the theory of finite groups, where the descending chain condition allows a natural choice of a distinguished tertiary decomposition and this yields a canonical decomposition of any finite group as a subdirect product of ''cotertiary'' finite groups. The group-theoretic application of the tertiary theory yields elementary structure theorems about Galois extensions of fields, where the tertiary decomposition of the Galois group transforms into a representation of a Galois extension as a compositum. For example, given a field F, there are distinguished ''tertiary'' field extensions of F, of which all other finite Galois extensions of F are compositums.