Some remarks on the formal power series ring
Matthew J. O’Malley · Bulletin de la Société mathématique de France · 1971
Let R be an integral domain with identity, let X be an indeterminate over jR, let S be the formal power series ring jR [[X]], and let G be a finite group of .R-automorphisms of S. If J? is a local ring (that is, a noetherian ring with unique maximal ideal M), and if R is complete in the M-adic topology, then P. SAMUEL shows the existence of f^S such that the ringThis paper was motivated by an attempt to generalize this result.Specifically, we will prove that the same conclusion holds if R is any noetherian integral domain with identity whose integral closure is a finite jR-module.In our efforts to obtain this result, we have had to make strong use of the results of [7] and theorem (2.6) of [6].The notion of topological completeness is essential.In paragraph 2, we develop the needed topological results and we make some additional comments concerning idealadic topologies under which R and R [[Xi, ..., Xn]] are complete.Paragraph 3 extends the result of Samuel.All rings in this paper are assumed to be commutative and, except for one brief mention in paragraph 2, to contain an identity element.The symbols GO and c^o are used throughout the paper to denote the sets of positive and nonnegative integers, respectively.The author wishes to express his sincere thanks to Professor Robert GILMER for his advice and encouragement during the preparation of this paper. Notation and terminology