Relative Vertex-Source-Pairs of Modules of and Idempotent Morita Equivalences of Rings

Here all rings have identities. Let R be a ring and let R -mod denote the additive category of left finitely generated R -modules. Note that if R is a noetherian ring, then R -mod is an abelian category and every R -module is a finite direct sum of indecomposable R -modules. Finite Group Modular Representation Theory concerns the study of left finitely generated O G -modules where G is a finite group and O is a complete discrete valuation ring with O / J ( O ) a field of prime characteristic p . Thus O G is a noetherian O -algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple O G -module) a theory of vertices and sources invariants. The vertices are derived from the set of p -subgroups of G . As suggested by the above, in Basic Definition and Main Results for Rings Section, let Σ be a fixed subset of subrings of the ring R and we develop a theory of Σ -vertices and sources for finitely generated R -modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R . For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that R = R e R . Set B = e R e so that B is a subring of R with identity e . Then, the functions e R ⊗ R ∗ : R − mod → B − mod and R e ⊗ B ∗ : B − mod → R − mod form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras.