On Mono-correct Modules

Let R be a commutative ring. It is well known that any artinian module is co-hopfian and any artinian module is mono-correct, but the converse is not true. Furthermore, commutative rings on which co-hopfian modules are artinian have been characterized. The aim of this work is to study the existence of commutative rings R on which mono-correct R-modules are artinian.
We establish that if there exists a commutative ring on which mono-correct R-modules are artinian, then it is an artinian ideal principal one. And on a non-zero commutative artinian principal ideal ring R, we have shown the existence of R-modules which are mono-correct but are not artinian. Hence a non-singleton unital commutative ring R such that every mono-correct R-module is artinian does not exist.