Anti-isomorphism of rings

2020 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]

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 $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).

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

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.

References

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