16B99-OppositeRing.tex

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\pmtitle{opposite ring}
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\begin{document}
If $R$ is a ring, then we may construct the {\it opposite ring} $R^{op}$ which has the same underlying abelian group structure, but with multiplication in the opposite order: the product of $r_1$ and $r_2$ in $R^{op}$ is $r_2 r_1$.
\par
If $M$ is a left $R$-module, then it can be made into a right $R^{op}$-module, where a module element $m$, when multiplied on the right by an element $r$ of $R^{op}$, yields the $rm$ that we have with our left $R$-module action on $M$.  Similarly, right $R$-modules can be made into left $R^{op}$-modules.
\par
If $R$ is a commutative ring, then it is equal to its own opposite ring.
\par
Similar constructions occur in the opposite group and opposite category.
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