Prime ideals in ultraproducts of commutative rings
Context: \(I\) is an infinite set and \(\mathcal U\) its ultrafilter. \(\{R_i\}_{i\in I}\) is a set of commutative rings; \(\prod_{\mathcal U}R_i\) denotes the ultraproduct mod \(\mathcal U\). The paper gives a number of consequences of a well-known Theorem of Łoś': If \(\phi\) is a sentence in the language of commutative rings, then \(\prod_{\mathcal U}R_i\) satisfies \(\phi\) if and only if \(R_i\) satisfies \(\phi\) for \(\mathcal U\)-many \(i\) (i.e. a property \(\mathcal S\) holds for \(\mathcal U\)-many \(i\), if the set of all \(i\) such that \(R_i\) satisfies \(\mathcal S\) is a member of \(\mathcal U\)). The goal of the paper is to describe classes of prime ideals of ultraproducts of commutative rings (such as Noetherian rings, Krull domains, QR-domains and finite character rings). If \(R\) is a commutative ring and \(V\) a non-empty set of prime ideals of \(R\), then denote \(\mathbf J(V)=\bigcap_{P\in V}P\). If \(\mathcal L\) is a sublattice of the lattice of all subsets of \(\mathcal P\subseteq \text{Spec}(R)\), for a proper filter \(\mathcal F\) define \((\mathcal F)=\bigcup_{V\in\mathcal F}\mathbf J(V)\) and if \(Q\) is a prime ideal of \(R\), then denote \(\mathcal F(Q)=\{ V\in\mathcal L:\mathbf J(V)\subseteq Q\}\). Then, several results establish equivalent conditions when the mappings \(\mathcal F\mapsto (\mathcal F)\) and \(Q\mapsto \mathcal F(Q)\) form bijective correspondences between filters and ideals (one such equivalent condition is that \(\mathcal L\) be saturated).
scientific article; zbMATH DE number 2157759
scientific article