Formal power series rings over a $\pi$-domain

Abstract

Let R be an integral domain, Χ be a set of indeterminates over R, and R\[\[Χ]]3 be the full ring of formal power series in Χ over R. We show that the Picard group of R\[\[Χ]]3 is isomorphic to the Picard group of R. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. An integral domain is a π-domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R\[\[Χ]]3 is a π-domain if R\[\[\_Χ\_1 , . . . , Χ\_\_n]] is a π-domain for every n ≥ 1. In particular, R\[\[Χ]]3 is a π-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 π-ring if every principal ideal is a product of prime ideals. We show that R\[\[Χ]]3 is a π-ring if R is a Noetherian regular ring.