Journal article
GCD and LCM-like identities for ideals in commutative rings
Journal of Algebra and its Applications, Vol.15(1), 1650010
2016
DOI: 10.1142/S0219498816500109
Abstract
Let A1,.,An(n ≥ 2) be ideals of a commutative ring R. Let G(k) (resp., L(k)) denote the product of all the sums (resp., intersections) of k of the ideals. Then we have L(n)G(2)G(4)G(2⌊ n/2⌋) ⊂ G(1)G(3) G(2⌈ n/2 ⌉-1). In the case R is an arithmetical ring we have equality. In the case R is a Prüfer ring, the equality holds if at least n-1 of the ideals A1,.,An are regular. In these two cases we also have G(n)L(2)L(4) L(2⌊ n/2 ⌋) = L(1)L(3) L(2⌈ n/2 ⌉-1). Related equalities are given for Prüfer v-multiplication domains and formulas relating GCD's and LCM's in a GCD domain generalizing gcd(a1, a2)lcm(a1, a2) = a1a2 are given. © 2016 World Scientific Publishing Company.
Details
- Title: Subtitle
- GCD and LCM-like identities for ideals in commutative rings
- Creators
- D.D. Anderson - University of IowaShuzo Izumi - Kindai UniversityYasuo Ohno - Tohoku UniversityManabu Ozaki - Waseda University
- Resource Type
- Journal article
- Publication Details
- Journal of Algebra and its Applications, Vol.15(1), 1650010
- Publisher
- World Scientific Publishing Co. Pte Ltd
- DOI
- 10.1142/S0219498816500109
- ISSN
- 0219-4988
- Language
- English
- Date published
- 2016
- Academic Unit
- Mathematics
- Record Identifier
- 9984230419702771
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