On representation of polynomial ring on a vector space via a linear transformation

Abstract

Listen

Translate article

A representation of a ring R is a ring homomorphism from R to the ring of all linear transformations from V to V (EndF (V)). From the field F, we can form a polynomial ring F[X]. A representation of F[X] is a ring homomorphism φ: F[X] → EndF (V ) via linear transformation T : V → V with φ(f(X)) = f(T) for all f(X) ∈ F[X]. For general ring representation ρ: R → EndF (V), we have notions of admissibility submodule, and completely reducible and simple admissible submodule. In this paper, we will show that admissible submodules of V are invariant under T, and a representation of F[X] is completely reducible.