Difference between revisions of "Anti-isomorphism of rings"
From Encyclopedia of Mathematics
(define anti-automorphism) |
(for example, conjugation of the quaternions) |
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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. | + | 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. |
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 $a + bi + cj + dk \mapsto a -bi -cj -dj$ 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
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