I need to solve this question:
Let $A=\begin{pmatrix} 1 &-2 & 1\\ 1& -2& 1\\ 1& -2& 1 \end{pmatrix}$.
Find the invariant elements and the degree of the $R$-module $M=\mathbb{C}^3$ where $R$ is the polynomial ring $R=\mathbb{C}[T]$ where $T(w)=Aw$.
I don't understand the defenitions in the question itself;
According to what I know a polynomial ring $\mathbb{C}[S]$ is of the form $\{\sum_{i=0}^nz_iS^i:n\in\mathbb{N},z_i\in\mathbb{C}\}$. So how should we look at $\mathbb{C}[T]$ for the $T$ we defined? How is $\mathbb{C}[T]$ a polynomial ring?
I will agree that the formulation of the question is a bit tangled. So here is my attempt at untangling this.
First of all, $R = \Bbb C[T]$, by itself, has nothing to do with any matrices. It's just a polynomial ring in one variable with complex coefficients.
Now, the question defines a $R$-module $M = \Bbb C^3$ by describing the multiplication map $R\times M\to M$. So, given a polynomial $f\in R$ and an element $w\in M$, what will $f\cdot w$ be?
If $f = k$ for some $k\in \Bbb C$ is a constant polynomial, then $f\cdot w$ is just $kw$. If we have $f = T$, then the question declares that $f\cdot w = Aw$, using standard matrix multiplication. This is enough to define the module multiplication entirely, because by the definition of modules, for a general $f$: $$ f = k_nT^n + k_{n-1}T^{n-1} + \cdots + k_1T + k_0 $$ we must have $$ f\cdot w = k_nA^nw + k_{n-1}A^{n-1}w + \cdot + k_1Aw + k_0w $$