\section{Ideals}
\label{section:banach-algebra-ideals}

\begin{definition}[Ideal]
\label{definition:banach-algebra-ideal}
    Let $A$ be a Banach algebra and $I \subset A$, then $I$ is a \textbf{left ideal} if:
    \begin{enumerate}
        \item $I$ is a subspace of $A$. 
        \item For each $x \in A$, $aI \subset I$. 
    \end{enumerate}

    and $I$ is a \textbf{two-sided ideal} if in addition to the above, 
    \begin{enumerate}[start=2]
        \item For each $x \in A$, $Ia \subset I$. 
    \end{enumerate}

    \textit{In this document, the "sidedness" of ideals are omitted. Within each block, the statement should hold as long as the interpretation is consistent. }
\end{definition}

\begin{definition}[Proper Ideal]
\label{definition:banach-algebra-proper-ideal}
    Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{proper} if $I \subsetneq A$.  
\end{definition}

\begin{definition}[Maximal Ideal]
\label{definition:maximal-ideal}
    Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{maximal} if:
    \begin{enumerate}
        \item $I$ is proper.
        \item For any proper ideal $J \subsetneq A$ with $J \supset I$, $I = J$. 
    \end{enumerate}

    The set $\cm(A)$ of maximal two-sided ideals of $A$ is the \textbf{maximal ideal space} of $A$. 
\end{definition}

\begin{proposition}
\label{proposition:ideal-gymnastics}
    Let $A$ be a unital Banach algebra and $I \subset A$ be a proper ideal, then:
    \begin{enumerate}
        \item $I$ contains no invertible elements. 
        \item $\ol I$ is also a proper ideal.
        \item $I$ is contained in a maximal ideal.
        \item If $I$ is maximal, then $I$ is closed. 
    \end{enumerate}
\end{proposition}
\begin{proof}[Proof, {{\cite{FollandHarmonic}}}. ]
    (1): If $I$ contains an invertible element, then $1 \in I$ and thus $A = I$. 

    (2): By \autoref{proposition:banach-algebra-inverse}, $G(A)$ is open. Since $G(A) \cap I = \emptyset$, $G(A) \cap \ol I = \emptyset$ as well. 

    (3): By Zorn's lemma. 

    (4): By (2). 
\end{proof}