16D20-Bimodule.tex

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\pmcreated{2013-03-22 12:01:18}
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\pmtitle{bimodule}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{16D20}
\pmsynonym{sub-bimodule}{Bimodule}
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\begin{document}
Let \fm{R} and \fm{S} be rings.  An \emph{(\fm{R},\fm{S})-bimodule} is
an abelian group \fm{M} which is a left module over \fm{R} and a right
module over \fm{S} such that the \fm{r}(\fm{ms})=(\fm{rm})\fm{s} holds
for each \fm{r} in \fm{R}, \fm{m} in \fm{M}, and \fm{s} in \fm{S}.
Equivalently, \fm{M} is an (\fm{R},\fm{S})-bimodule if it is a left
module over $R\otimes\opposite{S}$ or a right module over
$\opposite{R}\otimes S$.

When \fm{M} is an (\fm{R},\fm{S})-bimodule, we sometimes indicate this
by writing the module as ${}_RM_S$.

If \fm{P} is a subgroup of \fm{M} which is also an
(\fm{R},\fm{S})-bimodule, then \fm{P} is an
\emph{(\fm{R},\fm{S})-subbimodule} of \fm{M}.

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