anti-isomorphism

Let $R$ and $S$ be rings and $f:R⟶S$ 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 $m\ne n$, 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.

Title	anti-isomorphism
Canonical name	Antiisomorphism
Date of creation	2013-03-22 16:01:08
Last modified on	2013-03-22 16:01:08
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	15
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 13B10
Classification	msc 16B99
Defines	anti-endomorphism
Defines	anti-homomorphism
Defines	anti-isomorphic
Defines	anti-automorphism