Co-Hopfian Modules of Generalized Inverse Polynomials

Let R be an associative ring not necessarily possessing an identity and (S, ) a strictly totally ordered monoid which is also artinian and satisfies that 0 s for any sS. Assume that M is a left R-module having propertiy (F). It is shown that M is a co-Hopfian left R-module if and only if [M S, ] is a co-Hopfian left [[R S, ]]-module.