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Difference between revisions of "Anti-isomorphism of rings"

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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126401.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126402.png" /> into a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126403.png" /> that is an [[Isomorphism|isomorphism]] of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126404.png" /> onto the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126405.png" /> and for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126406.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012640/a0126407.png" />).
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{{TEX|done}}{{MSC|16}}
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''skew-isomorphism''
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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====
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* Igor R. Shafarevich, tr. M. Reid, ''Basic Notions of Algebra'', Springer (2006) ISBN 3-540-26474-4.  p.67

Revision as of 21:54, 29 November 2014

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
How to Cite This Entry:
Anti-isomorphism of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-isomorphism_of_rings&oldid=19259
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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