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# anti-isomorphism

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 $r_{{1}},r_{{2}}\in R$.

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

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

If $f$ is an anti-homomorphism and $R=S$ then $f$ is called an anti-endomorphism.

If $f$ is an anti-isomorphism and $R=S$ then $f$ 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 $M_{{m,n}}\to M_{{n,m}}$.

$R$ and $S$ 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.

## Mathematics Subject Classification

13B10*no label found*16B99

*no label found*

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## Corrections

Terminology by CWoo ✓

Not sure by CWoo ✓

anti-isomorphic by yark ✓

groups by Wkbj79 ✓