Generalized GCD rings (Q1591758)

Let \(R\) be a commutative ring with identity. An ideal \(I\) of \(R\) is called a multiplication ideal if every ideal contained in \(I\) is a multiple of \(I\). The authors define \(R\) to be a generalized greatest common divisor ring (G-GCD ring) if the intersection of any two finitely generated faithful multiplication ideals of \(R\) is also a finitely generated faithful multiplication ideal. \textit{D. D. Anderson} and \textit{D. F. Anderson} [Comment. Math. Univ. St. Pauli 28, 215-221 (1980; Zbl 0434.13001)] defined an integral domain to be a G-GCD domain if the intersection of any two invertible ideals is invertible. [\textit{S. Glaz}, Proc. Am. Math. Soc. 129, 2833-2843 (2001; see the following review Zbl 0971.13003) has also recently defined \(R\) to be a G-GCD ring if all principal ideals of \(R\) are projective and the intersection of any two finitely generated flat ideals of \(R\) is a finitely generated flat ideal.] The authors investigate various properties of G-GCD rings and generalize some results of \textit{J. Jäger} [Math.-Phys. Semesterber. Neue Folge 26, 230-243 (1979; Zbl 0415.13009)] and \textit{H. Lüneburg} [Ric. Mat. 38, 249-259 (1989; Zbl 0743.13011)] to finitely generated faithful multiplication ideals.