Dualities and the duality density theorems

Abstract

Dualities\({\langle,\, \rangle:S \times T \rightarrow K}\) for modules S = R S, T = TD and bimodule K = RKD over rings R, D are non-degenerate left dense K-pairings of S, T intertwined per adjoint, classification, and Galois correspondence theorems. Dualities are abundant per density theorems inspired by those of Jacobson and Chevalley. Duality theory generalizes classical duality theory and leads to a theory of duality semi-simplicity of rings R and R-modules S. The finite dimensional duality semi-simple algebras R are classified in terms of semi-simple algebras and bipolar algebras.