I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition
Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is finitely generated and free as an $R$-module. The trace of $x \in S$ is the trace of the $R$-linear endomorphism $m_x : S \rightarrow S$, $m_x(y)=xy$, and is denoted by $\operatorname{tr}(x)$ or $\operatorname{tr}_{S / R} (x)$. The norm of $x \in S$ is $\det(m_x)$, and is denoted by $\operatorname N_{S/R}(x)$ or $\operatorname N(x)$.
The word "understand" above is perhaps not optimal, I mean it's just a definition, so there isn't really something to understand, but this definition confuses me. I know from linear algebra how to compute the trace and the determinant, this is not the problem, but I have no idea how to determine this $R$-linear endomorphism. Could someone explain this to me?
I have also found on the internet the following exercise: Let $K := \mathbb{Q}$ and $L := \mathbb{Q}(\sqrt[3]{2})$ and we consider the element $\alpha = \sqrt[3]{2} + 3 \in L$. Write down the matrix that represents multiplication by $\alpha$ with respect to some basis of L as $K$-vectorspace. Determine $\operatorname N_{L / K}(\alpha)$ and $\operatorname{tr}_{L/K}(\alpha)$.
I'm not really sure, if my attempt of solving this is right: We can write down a basis of $L$ : $\{1, \sqrt[3]{2} , \sqrt[3]{2^2}\}$ and consider now multiplication by $\alpha$ :
$ \begin{eqnarray*} \alpha * (x + y \sqrt[3]{2} + z \sqrt[3]{4}) &=& (\sqrt[3]{2} + 3)* (x + y \sqrt[3]{2} + z \sqrt[3]{4}) \\ &=& 3x + 3y\sqrt[3]{2} +3z\sqrt[3]{4} +x\sqrt[3]{2} +y\sqrt[3]{4} +z\sqrt[3]{4}\sqrt[3]{2} \\ &=& 3x + 3y\sqrt[3]{2} +3z\sqrt[3]{4} +x\sqrt[3]{2} +y\sqrt[3]{4} +2z \\ &=& \begin{pmatrix} 3x+2z & 3y+x & 3z+y \\ \end{pmatrix} \begin{pmatrix} 1 \\ \sqrt[3]{2} \\ \sqrt[3]{4} \end{pmatrix} \end{eqnarray*} $
And now we can rewrite $\begin{pmatrix} 3x+2z \\ 3y+x \\ 3z+y \end{pmatrix}$ as
$ \begin{pmatrix} 3 & 0 & 2 \\ 1 & 3 & 0 \\ 0 & 1 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} $
So the matrix which represents multiplication by $\alpha$ would look like
$
\begin{pmatrix}
3 & 0 & 2 \\
1 & 3 & 0 \\
0 & 1 & 3
\end{pmatrix}
$
And we would have $\operatorname{tr}_{L/K}(\alpha) = 9$ and $\operatorname N_{L/K}(\alpha) = 27 + 2=29 $
Is this correct? And can I solve every exercise of this form like this, I mean for the arbitrary $x \in S$ in the definition above.