# 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.

Title anti-isomorphism Antiisomorphism 2013-03-22 16:01:08 2013-03-22 16:01:08 Mathprof (13753) Mathprof (13753) 15 Mathprof (13753) Definition msc 13B10 msc 16B99 anti-endomorphism anti-homomorphism anti-isomorphic anti-automorphism