Definition:R-Algebraic Structure Automorphism
Definition
Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ 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.
Also see
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.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules