Anti-isomorphism of rings

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 $a + bi + cj + dk \mapsto a -bi -cj -dj$ of the quaternion algebra.