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On Purity and Related Universal Properties of Extensions of Commutative Rings
並列摘要
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.
並列關鍵字
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延伸閱讀
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