# Formal power series rings over a <em>π</em>-domain

### Byung Gyun Kang

Pohang University of Science and Technology, South Korea### Dong Yeol Oh

National Institute for Mathematical Sciences, Daejeon, South Korea

## 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.

## Cite this article

Byung Gyun Kang, Dong Yeol Oh, Formal power series rings over a <em>π</em>-domain. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1429–1443

DOI 10.4171/JEMS/186