Let R ⊆ 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 {X1,...,X(subscript n)} over S and each ideal Ⅰ of R[X1,...,X(subscript n)], one has IS[X1,...,X(subscript n)] ∩ R[X1,...,X(subscript n)]=Ⅰ. Suppose also that Rüis a Prüfer domain. Then R is a pure R-submodule of S if and only if, for each unital homomorphism of commutative rings R→T, each chain of prime ideals of T can be covered by a corresponding chain of prime ideals of T⊗(subscript R) S.