AN ADDENDUM TO THE PAPER: MODULES WITH FINITELY MANY SUBMODULES

: "G. Picavet and M. Picavet-L'Hermitte, Modules with finitely many submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.": We characterize ring extensions R ⊂ S R⊂S having FCP (FIP), where S S is the idealization of some R R-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous.