| Version 10 |
Version 9 |
| Let $R$ and $S$ be rings and $f: R\longrightarrow S$ be a function such |
Let $R$ and $S$ 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 $r_{1}, r_{2} \in R$. |
that $f(r_{1}r_{2}) = f(r_{2})f(r_{1})$ for all $r_{1}, r_{2} \in R$. |
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Suppose that $R \neq S$. |
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| If $f$ is a homomorphism of the additive groups of $R$ and $S$, |
If $f$ is a homomorphism of the additive groups of $R$ and $S$, |
| then $f$ is called an {\it anti-homomorphsim}. |
then $f$ is called an {\it anti-homomorphsim}. |
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If $f$ is a bijection and anti-homomorphism,
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If $f$ is an isomorphism of the additive groups of $R$ and $S$,
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| then $f$ is called an {\it anti-isomorphism}. |
then $f$ is called an {\it anti-isomorphism}. |
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| If $f$ is an anti-homomorphism and $R=S$ |
Suppose $R=S$. |
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If $f$ is a homomorphism of the additive groups of $R$ and $S$, |
| then $f$ is called an {\it anti-endomorphism}. |
then $f$ is called an {\it anti-endomorphism}. |
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If $f$ is an anti-isomorphism and $R=S$
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If $f$ is an isomorphism of the additive groups of $R$ and $S$,
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| then $f$ is called an {\it anti-automorphism}. |
then $f$ is called an {\it anti-automorphism}. |
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| As an example, when $m \neq n$, the mapping that sends a matrix to its transpose |
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 |
(or to its conjugate transpose if the matrix is complex) is an anti-isomorphism |
| of $M_{m,n} \to M_{n,m}$. |
of $M_{m,n} \to M_{n,m}$. |
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