Hopf algebras of linear recurring sequences over rings and modules

The module of linear recurring sequences over a commutative ring $R$ can be considered as a Hopf algebra dual to the polynomial Hopf algebra over $R$. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and $k$-sequences.