GCD and LCM-like identities for ideals in commutative rings

D. D. Anderson, Shuzo Izumi, Yasuo Ohno, Manabu Ozaki

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.

Original languageEnglish
Article number1650010
JournalJournal of Algebra and its Applications
Volume15
Issue number1
DOIs
Publication statusPublished - 2016 Feb 1

Keywords

  • GCD
  • LCM
  • PVMD
  • Prüfer ring
  • arithmetical ring

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

Fingerprint Dive into the research topics of 'GCD and LCM-like identities for ideals in commutative rings'. Together they form a unique fingerprint.

  • Cite this