Fields of generalized power series
Let R be a commutative ring with unit element, let S be a commutative (multiplicatively written) semigroup with unit element, endowed with a compatible (partial) order relation \(\leq\). Let A be the set of all mappings \(f:\quad S\to R\) with support \(\sup p(f)=\{s\in S| \quad f(s)\neq 0\}\) which is artinian (it contains no infinite descending chain) and narrow (it contains no infinite subset of pairwise incomparable elements). With pointwise addition and convolution *, defined by \((f*g)(s)=\sum_{tu=s}f(t)g(u) \), it is known that A is a commutative ring with unit element. In this paper there are given the necessary and sufficient conditions for A to be a field, namely: (1) R is a field and S is a torsion free group; (2) either one of the following equivalent conditions holds: (a) for every \(s\in S\) there exists a positive integer k such that \(s^ k\leq 1\) or \(1\leq s^ k;\) (b) there exists a compatible total order \(\leq '\), finer than \(\leq\), such that \(1\leq 's\) if and only if there exists a positive integer k such that \(1\leq s^ k.\) The result was classical when the order \(\leq\) on S is a total order. The authors give an example of a field A obtained with this method, where the order \(\leq\) is only a partial order.
scientific article; zbMATH DE number 4106991