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Difference between revisions of "Anti-isomorphism of rings"

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(also skew-isomorphism, cite Shafarevich)
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''skew-isomorphism''
 
''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 $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).
 
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.
  
The opposite ring $R^{\mathrm{op}}$ is anti-isomorphic to $R$.
 
  
 
====References====
 
====References====
 
* Igor R. Shafarevich, tr. M. Reid, ''Basic Notions of Algebra'', Springer (2006) ISBN 3-540-26474-4.  p.67
 
* 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=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