GCD and LCM-like identities for ideals in commutative rings (Q2788755)

The authors are interested in studying ideal theoretic generalizations of properties related to well known properties of least common multiples and greatest common divisors in the integers. For instance, in the natural numbers (or any principal ideal domain), we have the following well known result gcd\((a_1, a_2)\)lcm\((a_1,a_2)=a_1a_2\). This has a natural generalization for ideals as \(((a_1)+(a_2))((a_1)\cap(a_2))=(a_1)(a_2)\) or since all ideals are principal in a PID, \((A_1+A_2)(A_1\cap A_2)=A_1A_2\). In fact, an integral domain \(R\) is a Prüfer domain if and only if \((A_1+A_2)(A_1\cap A_2)=A_1A_2\) holds for all ideals \(A_1\), \(A_2\) of \(R\).NEWLINENEWLINELet \(A_1, A_2, \ldots, A_n\) (\(n \geq 2\)) be \(n\) ideals in a commutative ring \(R\). Let \(G(k)\) (resp. \(L(k)\)) represent the product of all sums (resp. intersections) of \(k\) of the ideals. Then NEWLINE\[NEWLINEL(n)G(2)G(4) \cdots G(2\left \lfloor n/2 \right \rfloor \subseteq G(1)G(3) \cdots G(2\left \lfloor n/2 \right \rfloor -1).NEWLINE\]NEWLINE Equality holds if \(R\) is arithmetical, or if \(R\) is a Prüfer ring and at least \(n-1\) of the ideals are regular. In these cases, you also have the same equality with intersections and sums reversed: NEWLINE\[NEWLINEG(n)L(2)L(4) \cdots L(2\left \lfloor n/2 \right \rfloor =L(1)L(3) \cdots L(2\left \lfloor n/2 \right \rfloor -1).NEWLINE\]NEWLINE The authors then explore related equalities for Prüfer \(\nu\)-multiplication domains. They conclude the paper with some nice examples which help to illustrate many of the results from the paper.