The fields K) of generalized power series with coefficients in a field K and exponents in an additive abelian ordered group G play an important role in the study of real closed fields. The subrings K) consisting of series with non-positive exponents find applications in the study of models of weak axioms for arithmetic. Berarducci showed that the ideal JK) generated by the monomials with negative exponents is prime when is the additive group of the reals, and asked whether the (...) same holds for any G. We prove that this is the case and that in the quotient ring K)/J, each element admits at least one factorization into irreducibles. (shrink)

The field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays 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 ring K((G ≤ 0 )) of series (...) with non-positive exponents. Berarducci (see [1]) proved that K((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 that K((G ≤ 0 )) does have prime elements if G = (R, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup. (shrink)