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The purity of a completion

Published online by Cambridge University Press:  18 May 2009

S. H. Cox Jr
S. H. Cox Jr
Affiliation:
University of Puerto Rico, Rio Piedras, Puerto Rico 00931
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This note establishes two statements from R. M. Fossum's review [1] of a paper by E. A. Magarian [2]. Firstly, if A → B is a pure homomorphism (of commutative rings) then A[[x1,…,xs]] → B[[x1,…, xs]] is pure. Secondly, if Rn → R is a directed family of pure homomorphisms then ∪ Rn → R is pure. A consequence is that if Rn → R is a directed family of pure homomorphisms and if R is Noetherian, then ∪ Rn[[x1,…,xs]]is Noetherian.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 2 , July 1976 , pp. 127 - 129
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

REFERENCES

1.Fossum, R. M., Review number 290, Math. Reviews 48 (1974), 58–59.Google Scholar
2.Magarian, E. A., Direct limits of power series rings, Math. Scand. 31 (1972), 103–110.CrossRefGoogle Scholar
3.Gilmer, R. and Mott, J., Some results on contracted ideals, Duke Math. J. 37 (1970), 751–767.CrossRefGoogle Scholar
4.Enoch, E. E., On absolutely pure modules, Preprint, University of Kentucky.Google Scholar
5.Bourbaki, N., Algébre Commutative Chapitre 3 (Hermann, 1967).Google Scholar
6.Gilmer, R., Contracted ideals with respect to integral extensions, Duke Math. J. 34 (1967), 561–572.CrossRefGoogle Scholar
7.Gilmer, R., Contracted ideals in Krull domains, Duke Math. J. 37 (1970), 769–774.CrossRefGoogle Scholar