Ring Automorphisms

We have already defined Group Automorphisms. We now extend this concept to rings.

Definition: Let $(R, +, *)$ be a ring with multiplicative identity $1$. A function $\phi : R \to R$ is a Ring Automorphism if: 1) $\phi (a + b) = \phi (a) + \phi (b)$. 2) $\phi (a * b) = \phi (a) * \phi (b)$. 3) $\phi (1) = 1$. 4) $\phi$ is bijective. In other words, a ring automorphism of a ring $(R, +, *)$ is a bijective ring homomorphism from the ring onto itself.

Definition: Let $(R, +, *)$ be a ring. The set of all automorphisms on $R$ is denoted $\mathrm{Aut} (R)$.

It is easy to verify that $\mathrm{Aut} (R)$ is a group, and hence, $\mathrm{Aut} (R)$ is also sometimes called the Automorphism Group of $R$.