Maximal covers of chains of prime ideals

Abstract

Suppose \(f:S \rightarrow R\) is a ring homomorphism, with S and R not necessarily commutative and f not necessarily unitary. We study the connections between chains in \(\text {Spec} (S)\) and \(\text {Spec} (R)\). We focus on the properties lying over, incomparability, going down (GD), going up (GU) and strong going between (SGB). We prove necessary and sufficient conditions for f to satisfy each of the properties GD, GU and SGB, in terms of maximal \(\mathcal D\)-chains, where \({\mathcal {D}} \subseteq \text {Spec} (S)\) is a nonempty chain. We show that if f satisfies all of the above properties, then every maximal \(\mathcal D\)-chain is a maximal 1:1 cover of \(\mathcal D\). We also present an important example from quasi-valuation theory in which all of the above properties are satisfied. Moreover, we give equivalent conditions for the following property: for every chain \(\mathcal D \subseteq \text {Spec} (S)\) and for every maximal \(\mathcal D\)-chain \(\mathcal C \subseteq \text {Spec} (R)\), \(\mathcal C\) and \(\mathcal D\) are of the same cardinality.