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

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

    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Applied Mathematics

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