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[[\mathcal X]]_3 be the full ring of formal power series in \mathcal X] over R . We show that the Picard group of R[[\mathcal X]]_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[[\mathcal X]]_3 is a π -domain if R[[Χ_1 , . . . , Χ_n]] is a π -domain for every n ≥ 1 . In particular, R[[\mathcal X]]_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[[\mathcal X]]_3 is a π -ring if R is a Noetherian regular ring.