Hopf Ideals, Finite Duals, and Basic Constructions — Examples

Compute finite duals of polynomial algebras through finite quotient coefficients, finite-group duality, tensor Hopf structure, and Sweedler H4 with char(k)≠2. Contrast a coalgebra coideal, a ring ideal, and a Hopf ideal.

These are planned worked examples and counterexamples for hopf-ideals-finite-duals-and-basic-constructions, with empty item lists pending normal authoring and proof review. Each example must calculate from the stated hypotheses, and each counterexample must verify both its premises and the failed conclusion. This companion is a dependency leaf and supplies no theorem to another page.