On finite generation of R-subalgebras of R[X]

We show that over a complete discrete valuation ring R whose residue field is algebraically closed, any Noetherian R-subalgebra of R[X] is finitely generated and present examples of non-finitely generated Noetherian R-subalgebras of R[X] satisfying various properties. We also give a sufficient codimension-one criterion for a Noetherian R-subalgebra of R[X] to be finitely generated over R when R is locally factorial.