Generalized power series with a limited number of factorizations

Abstract

Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen–Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.