### Abstract

Let A_{1},.,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 A_{1},.,A_{n} 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(a_{1}, a_{2})lcm(a_{1}, a_{2}) = a_{1}a2 are given.

Original language | English |
---|---|

Article number | 1650010 |

Journal | Journal of Algebra and Its Applications |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Feb 1 |

### Fingerprint

### Keywords

- arithmetical ring
- GCD
- LCM
- Prüfer ring
- PVMD

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Journal of Algebra and Its Applications*,

*15*(1), [1650010]. https://doi.org/10.1142/S0219498816500109

**GCD and LCM-like identities for ideals in commutative rings.** / Anderson, D. D.; Izumi, Shuzo; Ohno, Yasuo; Ozaki, Manabu.

Research output: Contribution to journal › Article

*Journal of Algebra and Its Applications*, vol. 15, no. 1, 1650010. https://doi.org/10.1142/S0219498816500109

}

TY - JOUR

T1 - GCD and LCM-like identities for ideals in commutative rings

AU - Anderson, D. D.

AU - Izumi, Shuzo

AU - Ohno, Yasuo

AU - Ozaki, Manabu

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - arithmetical ring

KW - GCD

KW - LCM

KW - Prüfer ring

KW - PVMD

UR - http://www.scopus.com/inward/record.url?scp=84951844856&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951844856&partnerID=8YFLogxK

U2 - 10.1142/S0219498816500109

DO - 10.1142/S0219498816500109

M3 - Article

VL - 15

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

SN - 0219-4988

IS - 1

M1 - 1650010

ER -