Formal power series rings over a \(\pi \)-domain
Let \(R\) be an integral domain, \( \chi=\{X_{\lambda}\}_{\lambda\in \Lambda}\) an infinite set of indeterminates over \(R\) and \(S\) the weak direct sum of the additive semigroup \({\mathbb N}\) with itself \(|\Lambda|\) times. Let \(R\chi_3\) be the full ring of formal power series in \(\chi\) over \(R\), that is the set of maps \(f:S\rightarrow R\), with obvious addition and \((fg)(s)=\sum_{t+u=s} f(t)f(u)\), \(s\in S\), where \(t,u\) run in \(S\) such that \(t+u=s\). It shows that \(\mathrm{Pic } R\chi_3\cong\mathrm{Pic }R\). An integral domain is a \(\pi\)-domain if every principal ideal is a product of prime ideals, this happens exactly when it is locally an UFD and a Krull domain. It shows that \(R\chi_3\) is a \(\pi\)-domain if \(R[X_1,\ldots,X_n]\) is a \(\pi\)-domain for every \(n\geq 1\). Thus \(R\chi_3\) is a \(\pi\)-domain if \(R\) is a noetherian regular domain. Next these results are extended for non-domains.
scientific article; zbMATH DE number 5641379
scientific article