View source for Anti-isomorphism of rings

{{TEX|done}}{{MSC|16}}

''skew-isomorphism''

A mapping $\phi$ of a ring $A$ into a ring $B$ that is an [[isomorphism]] of the additive group of $A$ onto the additive group of $B$ and for which $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).

An ''anti-automorphism'' is an anti-isomorphism of a ring to itself: for example, the [[conjugation]] $a + bi + cj + dk \mapsto a -bi -cj -dk$ of the [[quaternion]] algebra.

The opposite ring $R^{\mathrm{op}}$ is anti-isomorphic to $R$.

====References====
* Igor R. Shafarevich, tr. M. Reid, ''Basic Notions of Algebra'', Springer (2006) ISBN 3-540-26474-4.  p.67