I’m currently learning about modules and am having trouble understanding something. It is claimed that if V is an F[X]-module (F a field) then V is an F-vector space together with a fixed linear endomorphism on V. Clearly V is an F-vector space, but the 2nd part of this claim is not so clear to me. It seems like V is an F-vector space together with a family of linear endomorphisms on V since each polynomial will induce a different linear transformation.
Thank you!
By the definition of the $F[X]$-module structure we have an $F$-linear (even $F[X]$-linear) map $$ T: V \rightarrow V, v\mapsto X\cdot v$$ Now any endomorphism induced by any polynomial acting on $V$ is completely determined by $T$. Indeed, if $f(X)=\sum_{j=0}^n a_j X^j$, then we have for $v\in V$ $$ f(X) \cdot v = f(T)(v).$$ Thus, specifying $T$ is enough.
It is a fun exercise to check the following: If we are given a $F[X]$-module $V$ s.t. it is finite dimensional as a $F$ vector space (with the vector space structure induced by the $F[X]$-module structure), then $V$ is free as a $F[X]$-module iff $V$ is the zero module. Hint: Cayley-Hamilton.