# Formal power series rings over a $π$-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, $X$ be a set of indeterminates over $R$, and $R[[X]]_{3}$ 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 $π$-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 $R[[X]]_{3}$ is a $π$-domain if $R[[X_{1},...,X_{n}]]$ is a $π$-domain for every $n≥1$. In particular, $R[[X]]_{3}$ 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 $R[[X]]_{3}$ 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 $π$-domain. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1429–1443

DOI 10.4171/JEMS/186