Opposite ring

Remarks

$R^{\mathrm{op}}$ is a ring with the same zero and the same unit element as $R$. Associativity holds because $(a * b) * c = c \cdot (b \cdot a) = (c \cdot b) \cdot a = a * (b * c)$. Example: for a commutative ring $K$, the transpose map $A \mapsto A^T$ is a ring isomorphism $M_n(K) \to M_n(K)^{\mathrm{op}}$, since $(AB)^T = B^T A^T = A^T * B^T$. Non-example: for a nonzero commutative ring $K$, the identity map of $M_2(K)$ is not a ring isomorphism $M_2(K) \to M_2(K)^{\mathrm{op}}$, because it would need $AB = BA$ for all $A, B$. A right $R$-module is the same thing as a left $R^{\mathrm{op}}$-module, with $r \cdot m = m r$.