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

Abstract

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 → EndR(M), with μ(r) := μr : M → M and μr(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→ EndR(M) and φ: R → EndR(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.