%
\label{prop:character-algebra-ring}
% Depends: II.D59, I.D21
$A_{\mathrm{spec}}(\mathbb{L})$ is a commutative ring under:
\begin{enumerate}
    \item \textbf{Pointwise addition.}
          $(\chi + \chi')(x) := \chi(x) + \chi'(x)$.
    \item \textbf{Pointwise multiplication.}
          $(\chi \cdot \chi')(x) := \chi(x) \cdot \chi'(x)$.
\end{enumerate}
The zero element is the character
$\chi_0(x) = 0$ for all $x$,
and the multiplicative identity
is the character $\chi_1(x) = 1$ for all $x$.

Pointwise addition and multiplication
are well-defined on ring homomorphisms:
if $\chi$ and $\chi'$ are ring homomorphisms,
then $\chi \cdot \chi'$ is multiplicative
(since $H_\tau^{\mathrm{cal}}$ is commutative).
The zero and identity characters
are the constant homomorphisms
$x \mapsto 0$ and $x \mapsto 1$ respectively.
Commutativity, associativity, and distributivity
follow from the corresponding properties
of $H_\tau^{\mathrm{cal}}$.