# Difference between revisions of "Anti-isomorphism of rings"

From Encyclopedia of Mathematics

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

## Revision as of 21:40, 29 November 2014

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.

**How to Cite This Entry:**

Anti-isomorphism of rings.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Anti-isomorphism_of_rings&oldid=35146

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article