On purity and related universal properties of extensions of commutative rings

Abstract

Let $R \subseteq S$ be a unital extension of commutative rings. Then $R$ is a pure $R$-submodule of $S$ if and only if, for each finite set of algebraically independent indeterminates $\{X_1, \, \dots \,,X_n\}$ over $S$ and each ideal $I$ of $R[X_1, \, \dots \,,X_n]$, one has $IS[X_1, \, \dots \,,X_n] \cap R[X_1, \, \dots \,,X_n]=I$. Suppose also that $R$ is a Pr\"ufer domain. Then $R$ is a pure $R$-submodule of $S$ if and only if, for each unital homomorphism of commutative rings $R \to T$, each chain of prime ideals of $T$ can be covered by a corresponding chain of prime ideals of $T \otimes_R S$.