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The Equality (A ∩ B)n = An ∩ Bn for Ideals

Published online by Cambridge University Press:  20 November 2018

Robert Gilmer  and
Anne Grams
Robert Gilmer
Affiliation:
Florida State University, Tallahassee, Florida
Anne Grams
Affiliation:
Florida State University, Tallahassee, Florida
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Let D be an integral domain with identity, and let R be a commutative ring. If n is a positive integer, R will be said to have property (n), (n)′, or (n)″ according as

property (n): For any x, y ∊ R, (x, y)n = (xn, yn).

property (n)′ : For any x ∊ R and any ideal A of R such that xn ∊ An, it follows that x ∊ A.

property (n)′ : For any ideals A, B of R, (A ∩ B)n = An ∩ Bn.

J. Ohm introduced property (n) in [7] in connection with the question: If n ≦ 2 and if D has property (n), must D be a Prüfer domain? (The integral domain D with identity is a Prüfer domain if each nonzero finitely generated ideal of D is invertible; equivalently, DP is a valuation ring for each proper prime ideal P of D.) Prior to Ohm's paper, it was known that if D has property (2) and if D is integrally closed, then D is Prüfer.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 792 - 798
Copyright
Copyright © Canadian Mathematical Society 1972

References

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