# Definition:R-Algebraic Structure Automorphism

## Definition

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ be an $R$-algebraic structure.

Let $\phi: S \to S$ be an $R$-algebraic structure isomorphism from $S$ to itself.

Then $\phi$ is an $R$-algebraic structure automorphism.

This definition continues to apply when $S$ is a module, and also when it is a vector space.

### Module Automorphism

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.

Let $\phi: G \to G$ be a module isomorphism to itself.

Then $\phi$ is a module automorphism.

### Vector Space Automorphism

Let $V$ be a $K$-vector space.

Let $\phi: V \to V$ be a vector space isomorphism to itself.

Then $\phi$ is a vector space automorphism.

## Linguistic Note

The word automorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus automorphism means self structure.