Definition:R-Algebraic Structure Endomorphism

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 homomorphism from $S$ to itself.

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

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

Also see

• Definition:Endomorphism

• Results about $R$-algebraic structure endomorphisms can be found here.

Linguistic Note

The word endomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix endo- (from ἔνδον') meaning inner or internal.

Thus endomorphism means internal structure.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations