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Going Down and Open Extensions

Published online by Cambridge University Press:  20 November 2018

Stephen McAdam
Stephen McAdam*
Affiliation:
University of Texas, Austin, Texas
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We call an extension of commutative rings, R ⊂ T, open if the spec mapping from spec (T) to spec (R), which sends the prime Q of T to Q ∩ R, is an open mapping. It is easy to show, as for example in [1], that if R ⊂ T is open then it satisfies going down. In general, the converse is false, as is shown by Z ⊂ (2z) with Z the integers. To the best of this author's knowledge, it is an open question whether for an integral extension, going down and open are equivalent.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 111 - 114
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Ferrand, D., Morphisms Entiers Universellement Overt (Manuscript).Google Scholar
2. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, 1970).Google Scholar
3. Matsumura, H., Commutative algebra (Benjamin, New York, 1970).Google Scholar