# Definition:Endomorphism

## Definition

An endomorphism is a homomorphism from an algebraic structure into itself.

### Semigroup Endomorphism

Let $\left({S, \circ}\right)$ be a semigroups.

Let $\phi: S \to S$ be a (semigroup) homomorphism from $S$ to itself.

Then $\phi$ is a semigroup endomorphism.

### Group Endomorphism

Let $\struct {G, \circ}$ be a group.

Let $\phi: G \to G$ be a (group) homomorphism from $G$ to itself.

Then $\phi$ is a group endomorphism.

### Ring Endomorphism

Let $\struct {R, +, \circ}$ be a ring.

Let $\phi: R \to R$ be a (ring) homomorphism from $R$ to itself.

Then $\phi$ is a ring endomorphism.

### Field Endomorphism

Let $\struct {F, +, \circ}$ be a field.

Let $\phi: F \to F$ be a (field) homomorphism from $F$ to itself.

Then $\phi$ is a field endomorphism.

### $R$-Algebraic Structure Endomorphism

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.

### Module Endomorphism

Let $R$ be a ring.

Let $M$ be an $R$-module.

A module endomorphism of $M$ is a module homomorphism from $M$ to itself.

## 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.