On a ring associated to F[x].

For a field F and the polynomial ring F [x] in a single indeterminate, we define Ḟ [x] = {α ∈ End_F(F [x]) : α(ƒ) ∈ ƒF [x] for all ƒ ∈ F [x]}. Then Ḟ [x] is naturally isomorphic to F [x] if and only if F is infinite. If F is finite, then Ḟ [x] has cardinality continuum. We study the ring Ḟ[x] for finite fields F. For the case that F is finite, we discuss many properties and the structure of Ḟ [x].