Open AccessOn purity and related universal properties of extensions of commutative rings
Author(s) -
David E. Dobbs
Publication year - 2010
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.41.2010.723
Subject(s) - mathematics , unital , commutative property , homomorphism , prime (order theory) , commutative ring , ideal (ethics) , domain (mathematical analysis) , pure mathematics , discrete mathematics , combinatorics , algebra over a field , mathematical analysis , law , political science
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 $T otimes_R S$