maximal idealMaximalIdeal2001-10-20 01:43:562002-04-20 00:02:21Definition\usepackage{amssymb}
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\usepackage{xypic}Let $R$ be a ring with identity. A proper left (right, two-sided) ideal $\mathfrak{m} \subsetneq R$ is said to be {\em maximal} if $\mathfrak{m}$ is not a proper subset of any other proper left (right, two-sided) ideal of $R$.
One can prove:
\begin{itemize}
\item A left ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple left $R$-module.
\item A right ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple right $R$-module.
\item A two-sided ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple ring.
\end{itemize}
All maximal ideals are prime ideals. If $R$ is commutative, an ideal $\mathfrak{m} \subset R$ is maximal if and only if the quotient ring $R/\mathfrak{m}$ is a field.