Existence of prime elements in rings of generalized power series

Abstract

AbstractThe fieldK((G)) of generalized power series with coefficients in the fieldKof characteristic 0 and exponents in the ordered additive abelian groupGplays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ringK((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved thatK((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove thatK((G≤0)) does have prime elements ifG= (ℝ, +) is the additive group of the reals, or more generally ifGcontains a maximal proper convex subgroup.