On Hasse–Schmidt rings and module algebras

Recently, Moosa and Scanlon introduced (iterative) Hasse–Schmidt systems D and, given such a Hasse–Schmidt system, they defined (iterative) D-rings, generalizing rings with higher derivation as introduced by Hasse and Schmidt in 1937. We show that there is a bijection between Hasse–Schmidt systems D and cocommutative coalgebras D. For a given Hasse–Schmidt system D with associated coalgebra D we show that D-rings are in bijection to algebras with a D-measuring on them. Under these correspondences iterative Hasse–Schmidt systems D correspond to cocommutative bialgebras D and iterative D-rings correspond to D-module algebras.