# Definition:Endomorphism

## Definition

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

### Semigroup Endomorphism

Let $\struct {S, \circ}$ 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.

## Also known as

Some sources define an **endomorphism** in the same way that $\mathsf{Pr} \infty \mathsf{fWiki}$ defines a self-map.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the word for a self-map on an algebraic structure which is at the same time a homomorphism.

## Also see

- Definition:Homomorphism (Abstract Algebra)
- Definition:Monomorphism (Abstract Algebra)
- Definition:Epimorphism (Abstract Algebra)
- Definition:Isomorphism (Abstract Algebra)

- Results about
**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 $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras