On purity and related universal properties of extensions of commutative rings

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 $R o T$, each chain of prime ideals of $T$ can be covered by a corresponding chain of prime ideals of $T otimes_R S$