Difference between revisions of "Anti-isomorphism of rings"
From Encyclopedia of Mathematics
(also skew-isomorphism, cite Shafarevich) |
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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 $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$). | 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 $R$. | ||
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. | 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==== | ====References==== | ||
| − | * Igor R. Shafarevich, tr. M. Reid, ''Basic Notions of Algebra'', Springer (2006) ISBN 3-540-26474-4 | + | * Igor R. Shafarevich, tr. M. Reid, ''Basic Notions of Algebra'', Springer (2006) {{ISBN|3-540-26474-4}} p.67 |
Latest revision as of 19:11, 24 November 2023
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=35152
Anti-isomorphism of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-isomorphism_of_rings&oldid=35152
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article