Infinite-Order Differential Operators in the Module of Formal Generalized Functions and in a Ring of Formal Power Series

We obtain the general form of continuous linear mappings acting in the module of formal generalized functions over a commutative ring and commuting with the operator of differentiation (or shift). We also prove that a continuous linear mapping acting in the ring of formal power series over the valuation ring of a complete non-Archimedean field and commuting with the operator of differentiation is a differential operator of infinite order.