PlanetMath: anti-isomorphism

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anti-isomorphism

(Definition)

Let and be rings and $f: R\longrightarrow S$ be a function such that $f(r_{1}r_{2}) = f(r_{2})f(r_{1})$ for all $r_{1}, r_{2} \in R$ .

If is a homomorphism of the additive groups of and , then is called an anti-homomorphsim.

If is a bijection and anti-homomorphism, then is called an anti-isomorphism.

If is an anti-homomorphism and then is called an anti-endomorphism.

If is an anti-isomorphism and then is called an anti-automorphism.

As an example, when $m \neq n$ , the mapping that sends a matrix to its transpose (or to its conjugate transpose if the matrix is complex) is an anti-isomorphism of $M_{m,n} \to M_{n,m}$ .

and are anti-isomorphic if there is an anti-isomorphism $R \to S$ .

All of the things defined in this entry are also defined for groups.

"anti-isomorphism" is owned by Mathprof.

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Also defines:

anti-endomorphism, anti-homomorphism, anti-isomorphic, anti-automorphism

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Cross-references: groups, complex, conjugate transpose, transpose, matrix, mapping, bijection, additive groups, homomorphism, function, rings
There are 7 references to this entry.

This is version 12 of anti-isomorphism, born on 2006-06-17, modified 2007-06-01.
Object id is 8057, canonical name is AntiIsomorphism.
Accessed 3157 times total.

Classification:

AMS MSC:	16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous)
	13B10 (Commutative rings and algebras :: Ring extensions and related topics :: Morphisms)

Pending Errata and Addenda

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