Duality and products in algebraic (co)homology theories

The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a ${\times }_A$-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie-Rinehart (co)homology.