# Anti-isomorphism of rings

From Encyclopedia of Mathematics

(Redirected from Anti-automorphism)

2010 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

**How to Cite This Entry:**

Anti-automorphism.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Anti-automorphism&oldid=39390