Co-Hopfian Modules

If $R$ is a commutative ring with $1$, then a co-Hopfian $R$-module $M$ is necessarily a module of fractions over a ring of fractions. For $R$ commutative Noetherian, we use Matlis’ notion of modules with maximal orders together with localization to obtain results about co-Hopfian submodules of indecomposable injectives and co-Hopfian finitely generated P-primary $R$-modules. With suitable restriction, we determine co-Hopfian and Hopfian submodules of a co-Hopfian injective module.