TITLE: Name for the module corresponding to a square matrix
QUESTION [5 upvotes]: I recently learned that for each $n \times n$ matrix $A$ with entries in some field $F$, there is a corresponding $F[x]$-module $M_A$. Namely, $M_A$ is the set $F^n$ with vector addition defined as usual, but with scalar multiplication defined by $f(x) \cdot v = f(A) \, v$ for each $f(x) \in F[x]$. My question is whether there is a name for the module $M_A$, and if so, what is it?
I'm sorry that this is a bit of a trivial question, but I've looked around and have been unable to find an answer. If this question is inappropriate for this site please don't hesitate to let me know. Thanks.

REPLY [2 votes]: One thing we can say is that the map $\Phi_A: \Bbb F[x] \to \Bbb F^{n \times n}$ given by $\Phi_A(f) = f(A)$ is an algebra homomorphism.  In fact, we might be able to say some thing like $\Phi_A$ is the natural homomorphism induced by $A$.  With representation theory in mind, we might call this the representation of $\Bbb F[x]$ induced by $A$.
Having said all of that, $f(x)\cdot v$ might be called the multiplication induced by the representation $\Phi_A$.  The resulting module is probably best referred to as the image of this representation.