On representation of a ring on a free module over a commutative ring with identity

Let R be a commutative ring with identity and M be a free R -module then we always have a representation of R , that is homomorphism ring μ : R → End R ( M ), with μ ( r ) := μ r : M → M and μ r ( m ) = rm for all r ∈ R and for all m ∈ M . In this paper, we will present some properties of representations of ring R on R -module, based on some notions in representation of R on vector space, such as admissible submodule, equivalence of two representations, decomposable representation and completely reducible representation. It will be shown that if M , N are two free R -modules then two representations μ : R → End R ( M ) and φ : R → End R ( N ) are equivalent if and only if there is a module isomorphism T : M → N . If R is a principle ideal domain(PID), then it will be shown that every submodule of M is an admissible submodule of M , every representation of ring R on a free R -module is decomposable, and a representation of R on M is completely reducible if and only if M is semisimple.