opposite ring OppositeRing 2001-10-20 03:20:05 2004-03-11 00:37:16 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} 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.