Algebraic properties of rings of generalized power series (Q1602848)

eummary: The fields \(K((G))\) 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 \(K((G^{\leq 0}))\) consisting of series with nonpositive exponents find applications in the study of models of weak axioms for arithmetic. \textit{A. Berarducci} [Trans. Am. Math. Soc. 352, 553-577 (2000; Zbl 0957.13020)] showed that the ideal \(J\subseteq K((G^{\leq 0}))\) generated by the monomials with negative exponents is prime when \(G= (\mathbb{R},+)\) 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 \(K((G^{\leq 0}))/J\), each element (not in \(K\)) admits at least one factorization into irreducibles.