anti-isomorphism

Let $R$ and $S$ be rings and $f:R⟶S$ be a function suchthat $f(r_1{r}_2)=f(r_2)f(r_1)$ 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-isomorphismof $M_{m,n}\to M_{n,m}$.

$R$ 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.