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

Abstract

Let $R$ be an integral domain, $X$ be a set of indeterminates over $R$, and $R[[\mathcal{X}]{]}_3$ 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 $\pi$-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 $R[[\mathcal{X}]{]}_3$ is a $\pi$-domain if $R[[X_1,...,X_n]]$ is a $\pi$-domain for every $n\ge 1$. In particular, $R[[\mathcal{X}]{]}_3$ 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 $R[[\mathcal{X}]{]}_3$ is a $\pi$-ring if $R$ is a Noetherian regular ring.