On modules with DICC
A module M over a commutative ring R with unit is said to be DICC when it contains no doubly infinite chain of submodules and SDICC when it is DICC but neither ACC nor DCC. It is well-known that the existence of a faithful ACC module implies R is ACC, and a result of \textit{W. Heinzer} and \textit{D. Lantz} [J. Algebra 95, 201-216 (1985; Zbl 0588.13015)] states that, for R complete and quasi-local, the existence of a faithful DCC module implies R is ACC. The author gives an example to show that the existence of a faithful SDICC module does not imply R is DICC even when R is quasi-local and R/\({\mathfrak n}\) is complete (\({\mathfrak n}=rad 0)\). However the existence of a faithful SDICC module does imply R is weakly DICC, i.e. R/\({\mathfrak n}\) is Noetherian, \({\mathfrak n}\) is DCC and nilpotent. The author also adds to her characterizations of DICC rings and modules [J. Algebra 101, 489-496 (1986; Zbl 0589.13009) and 105, 429-436 (1987; Zbl 0606.13013)] a new characterization of SDICC.
scientific article; zbMATH DE number 3987422