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
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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
-modules where
G is a finite group and
is a complete discrete valuation ring with
a field of prime characteristic
p. Thus
is a noetherian
-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
-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
be a fixed subset of subrings of the ring
R and we develop a theory of
-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
. Set
so that
B is a subring of
R with identity
e. Then, the functions
and
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.
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