\documentclass{article}
\usepackage{amsmath,amssymb,geometry}
\geometry{a4paper, margin=0.85in}

\title{Cheat Sheet: Finite Ring with Unique Zero Divisor}
\date{}

\begin{document}
\maketitle

\section*{Given}
$R$ finite commutative ring with $1\neq 0$, and exactly one nonzero zero divisor.

\section*{Goal}
Show $|R|=4$ and classify $R$.

\section*{Algorithm}

\textbf{1. Isolate element}\\
Let $x\neq 0$ be the unique zero divisor.

\textbf{2. Force nilpotency}\\
Since $x$ is a zero divisor, $\exists y\neq 0$ with $xy=0$.  
Then $y$ is a zero divisor $\Rightarrow y=x$. Hence:
\[
x^2=0.
\]

\textbf{3. Control units}\\
Let $u\in R^\times$. Then $ux\neq 0$ and
\[
(ux)x=u x^2=0 \Rightarrow ux \text{ is zero divisor}.
\]
By uniqueness: $ux=x \Rightarrow (u-1)x=0$.

\textbf{4. Structure collapse}\\
Elements are forced into:
\[
R=\{0,1,x,1+x\} \Rightarrow |R|=4.
\]

\textbf{5. Classification}\\
Since $x^2=0$ and $x\neq 0$, $R$ is a local ring of order $4$:
\[
R \cong \mathbb{Z}_4 \quad \text{or} \quad \mathbb{F}_2[X]/(X^2).
\]

Distinguish by characteristic:
\[
\mathrm{char}(R)=2 \Rightarrow \mathbb{F}_2[X]/(X^2), 
\quad
\mathrm{char}(R)=4 \Rightarrow \mathbb{Z}_4.
\]

\section*{Triggers}
- uniqueness of zero divisor $\Rightarrow$ collapse argument  
- finite ring $\Rightarrow$ unit/zero divisor dichotomy  
- multiplication by units preserves structure  

\end{document}