\section{Characteristic of a Ring}

\begin{definition}[Characteristic of a Ring]
	The \textit{characteristic} of a ring $R$ is the least positive integer $n$ such that $nx = 0$ for all $x$ in $R$. If no such integer exists, we say that $R$ has characteristic 0. The characteristic of $R$ is denoted by $\characteristic R$.
\end{definition}

\begin{theorem}[Characteristic of a Ring with Unity]
	Let $R$ be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of $R$ is 0. If 1 has order $n$ under addition, then the characteristic of $R$ is $n$.
\end{theorem}

\begin{theorem}[Characteristic of an Integral Domain]
	The characteristic of an integral domain is 0 or prime.
\end{theorem}