Prime ideals in infinite products of commutative rings (Q6577576)

Let \(\Lambda\) be a set, let \((D_\lambda)_{\lambda\in\Lambda}\) be a family of commutative rings and let \(\max(D_\lambda)\) denote the set of all maximal ideals of \(D_\lambda\). Further, let \(\mathcal B=\prod_{\lambda\in\Lambda}\mathcal P(\max(D_\lambda))\) be the product of the Boolean algebras \((\mathcal P(\max(D_\lambda)),\cap,\cup)\) where \(\mathcal P(-)\) denotes the power set.\N\NIn this paper, the authors describe the prime ideals and the maximal ideals of the product \(\mathcal R=\prod_{\lambda\in\Lambda}D_\lambda\) of the rings \(D_\lambda\). They show that every maximal ideal of \(\mathcal R\) is induced by an ultrafilter on the Boolean algebra \(\mathcal B\) and use it to characterize the maximal ideals of \(\mathcal R\) when every \(D_\lambda\) is in a certain class of rings containing all finite character domains and all one-dimensional domains. They also give a complete characterization of the prime ideals of \(\mathcal R\) when every \(D_\lambda\) is a Prüfer domain.