The property of having a finite number of orbits under the regular action has been used to study properties of rings and algebras. For example, in ring theory, Yasuyuki Hirano was able to use this property to show that rings with finitely many orbits under the regular action can be decomposed as direct sum of uniserial rings and a finite ring. In this thesis, we study modules under the regular action. More precisely, if R is a unital ring and M is a left(right) R-module, we describe all modules M that have finitely many orbits under the regular action. Along the way, we give a (new) module theoretical proof to the theorem of Yasuyuki Hirano on the classification of rings with finitely many orbits under the regular action which was proven using using methods from ring theory. Our charaterization of modules with finitely many orbits under the regular action shows a connection between algebras with finitely many submodules and distributive modules. A particular algebra that is of interest to us is the incidence algebra of a finite poset.
Incidence algebras were originally introduced in the 1960's by Gian-Carlo Rota as a way to study combinatorial problems but it became apparent later on that such algebras were an interesting object to study in their own right. They include ring theoretical examples such as the product of copies of a ring R and the upper triangular matrices over R. Robert B. Feinberg in his work on incidence algebras developed an internal characterization of incidence algebras of lower finite quasi-ordered sets. For example, he showed that an associative unital complete topological algebra Λ over a field K, where K has the discrete topology, is isomorphic to an incidence algebra if and only if
1. Λ has a faithful unital left module M with a distributive lattice of submodules. Further, every finitely generated submodule of M is finite dimensional and Λ has the coarset topology such that its action on M is continuous in Λ, when M has the discrete topology.
2. For every maximal closed ideal J, Λ/J is isomorphic to Mn(k) for some integer n.
3. For every closed ideal J, the center Λ/J is isomorphic to the direct product of copies of k.
This thesis investigates the deformations of incidence algebras and how such deformations relate to cohomology. We show that distributivity of projective indecomposable modules of algebras largely characterizes precisely those algebras which are deformations of incidence algebras