Summary: Using the standard duality we construct a linear embedding of an associated module for a pair of ideals in an extension of a Dedekind ring into a tensor square of its fraction field. Using this map we investigate properties of the coefficient-wise multiplication on associated orders and modules of ideals. This technique allows to study the question of determining when the ring of integers is free over its associated order. We answer this question for an Abelian totally wildly ramified $p$-extension of complete discrete valuation fields whose different is generated by an element of the base field. We also determine when the ring of integers is free over a Hopf order as a Galois module.

11S15, 11S20, 11S31

complete discrete valuation (local) fields, additive Galois modules, formal groups, associated Galois modules