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Unverified Commit 34e38d0e authored by rrueger's avatar rrueger
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Algebra I: Modules. Remark on left and right modules for abelian rings

parent ae93a456
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\item $m1 = m$.
\end{enumerate}
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In the case of a commutative ring $R$ the distinction between left and right
$R$-modules becomes obsolete.
\begin{remark}
In the case of a commutative ring $R$ the distinction between left and right
$R$-modules becomes obsolete. This fact is not immediately obvious, and
stems from how the action $ax$ is actually defined. A full discussion
pertaining to this issue is outside the scope of this summary, but the
subject of pages 157--164 of ``Basic Algebra I:\ Second Edition'' by Nathan
Jacobson. In short, beginning with a left $R$-module and defining $xa := ax$
gives rise to a new $R^o$-module. Here, $R^o$ denotes the opposite ring of
$R$, where multiplication is reversed. In the case of a commutative ring $R
= R^o$ and thus the left $R$-module is equal to the right $R$-module.
\end{remark}
\end{definition}
\begin{definition}
......
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