14 lines
457 B
TeX
14 lines
457 B
TeX
\section{Ideals}
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\begin{definition}[Ideal]
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A subring $A$ of a ring $R$ is called a (two-sided) \textit{ideal} of $R$ if for every $r \in R$ and every $a \in A$ both $ra$ and $ar$ are in $A$.
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\end{definition}
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\begin{theorem}[Ideal Test]
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A nonempty subset $A$ of a ring $R$ is an ideal of $R$ if
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\begin{enumerate}
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\item $a-b \in A$ whenever $a,b \in A$.
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\item $ra$ and $ar$ are in $A$ whenever $a \in A$ and $r \in R$.
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\end{enumerate}
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\end{theorem}
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