= Opposite ring
{title2=$R^{\mathrm{op}}$}
The opposite ring has the same additive group and identity as a <ring> $R$, with multiplication $a\cdot_{\mathrm{op}}b=ba$. Left <modules> over it are right <modules> over $R$. Forming the opposite exchanges left and right chain conditions and reverses the order of composition when interpreting endomorphisms as right multiplications. For an <associative algebra> with its scalar action retained, this construction gives the <opposite algebra>.
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