16D25-ProductOfLeftAndRightIdeal.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ProductOfLeftAndRightIdeal}
\pmcreated{2013-03-22 17:38:09}
\pmmodified{2013-03-22 17:38:09}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{product of left and right ideal}
\pmrecord{7}{40057}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{16D25}
\pmrelated{ProductOfIdeals}
\pmrelated{Intersection}
\pmrelated{IdealMultiplicationLaws}

\endmetadata

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\begin{document}
Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of a ring $R$.\, Denote by\, $\mathfrak{ab}$\, the subset of $R$ formed by all finite sums of products $ab$ with\, $a \in \mathfrak{a}$\, and\, 
$b \in \mathfrak{b}$.\, It is straightforward to verify the following facts:
\begin{itemize}
\item If $\mathfrak{a}$ is a \PMlinkname{left}{Ideal} and $\mathfrak{b}$ a right ideal, $\mathfrak{ab}$\, is a two-sided ideal of $R$.
\item If both $\mathfrak{a}$ and $\mathfrak{b}$ are two-sided ideals, then\,
 $\mathfrak{ab} \subseteq \mathfrak{a}\cap\mathfrak{b}$.
\end{itemize}
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\end{document}