# Definition:R-Algebraic Structure Endomorphism

## Contents

## 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

## 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): $\S 28$