Modules (co)reduced relative to another module

Let R be a commutative unital ring, a an ideal of R and M a fixed R-module. We introduce and study generalisations of a-reduced modules, Rₐ and a-coreduced modules, Cₐ studied in the literature. They are called modules a-reduced with respect to M, RMₐ and modules a-coreduced with respect to M, CMₐ. The paper lifts several known results about Rₐ and Cₐ to larger classes of modules RMₐ and CMₐ respectively. Results about Rₐ and Cₐ can be retrieved by taking $M= R$.