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.