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

From Encyclopedia of Mathematics

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− | + | ''skew-isomorphism'' | |

− | An ''anti-automorphism'' is an anti-isomorphism of a ring to itself. | + | 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 $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$). |

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

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+ | 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. | ||

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+ | ====References==== | ||

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

## Latest revision as of 21:54, 29 November 2014

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-isomorphism of rings.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Anti-isomorphism_of_rings&oldid=35143

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