On finite generation of \(R\)-subalgebras of \(R[X]\) (Q947508)

It is well-known that polynomial rings over a discrete valuation ring \(R\) may have non-noetherian, hence not finitely generated, \(R\)-subalgebras. Even noetherian \(R\)-subalgebras may have no finite generator systems as shown by \textit{P. Eakin} [Proc. Am. Math. Soc. 31, 75--80 (1972; Zbl 0232.16009)]. The authors study conditions under which every noetherian \(R\)-subalgebra is finitely generated, and show that this holds in the case, when \(R\) is a complete discrete valuation ring whose residue class field \(k\) has algebraical closure having finite degree over \(k\). They produce also several examples of rings \(R\) such that \(R[X]\) has noetherian \(R\)-subalgebras which are not finitely generated.