Formal power series rings over a pi-domain

Abstract

Let R be an integral domain, X be a set of indeterminates over R, and R[[1X]](3) be the full ring of formal power series in X over R. We show that the Picard group of R[[X]](3) is isomorphic to the Picard group of R. An integral domain is called a pi-domain if every principal ideal is a product of prime ideals. An integral domain is a pi-domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R[[X]](3) is a pi-domain if R[[X-1,...,X-n]] is a pi-domain for every n >= 1. In particular, R[[X]](3) is a pi-domain if R is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring R with identity is called a pi-ring if every principal ideal is a product of prime ideals. We show that R[[X]](3) is a pi-ring if R is a Noetherian regular ring.