# Modules, Polynomial Rings, Finite Dimensional Vector Space and Multiplication

Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

Â© BrainMass Inc. brainmass.com March 4, 2021, 6:23 pm ad1c9bdddfhttps://brainmass.com/math/ring-theory/polynomial-rings-finite-dimensional-vector-space-42503

#### Solution Preview

This is a Linear Algebra Problem

--------------------------------------------------------------------------------

Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

***********************************************************************

Definition (from Fraleigh)

Let R be a ring. A (left) R-module consists of an abelian group M together with an operation of external multiplication of each element of M by each element of R on the left such that for all a, b elements of M and r, s elements of R, the following conditions are satisfied

(1) ra is an element of M (2) r(a + b) = ra + rb (3) (r + s)a = ra + sa (4) (rs)a = r(sa)

Fraleigh notes that an R-module is very much like a vector space except that the scalars (elements of R) need only form a ring.

Alternate definition (from Jacobson)

Consider a homomorphism of a given ring R onto the ring of endomorphisms, End(M), of an abelian group M. If n is such a homomorphism, r in R, n(r) is an element of EndM, ...

#### Solution Summary

Modules, Polynomial Rings, Finite Dimensional Vector Space and Multiplication are investigated. The solution is detailed and well presented.