GCD and LCM-like identities for ideals in commutative rings

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 [Formula: see text] 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 [Formula: see text] 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.