Let R be a commutative ring and M be an unital R-module. M is called co-Hopfian if any injective endomorphism of M is an isomorphism, M is called weakly co-Hopfian if any injective endomorphism of M is essential. The ring R is called weakly FGD-ring if any weakly co-Hopfian R-module has finite uniform dimension. In this note, we prove that a commutative ring R, on which every uniform R-module contains a simple submodule, is a weakly FGD-ring if and only if it is an Artinian principal ideal ring.

weakly FGD-rings, Artinian principal ideal rings, co-Hopfian module, weakly co-Hopfian module, uniform dimension.