# Definition:Epimorphism (Abstract Algebra)

## Definition

A homomorphism which is a surjection is described as epic, or called an epimorphism.

### Semigroup Epimorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups.

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

Then $\phi$ is a semigroup epimorphism if and only if $\phi$ is a surjection.

### Group Epimorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

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

Then $\phi$ is a group epimorphism if and only if $\phi$ is a surjection.

### Ring Epimorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.

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

Then $\phi$ is a ring epimorphism if and only if $\phi$ is a surjection.

### Field Epimorphism

Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields.

Let $\phi: R \to S$ be a (field) homomorphism.

Then $\phi$ is a field epimorphism if and only if $\phi$ is a surjection.

### $R$-Algebraic Structure Epimorphism

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures.

Then $\phi: S \to T$ is an $R$-algebraic structure epimorphism if and only if:

$(1): \quad \phi$ is a surjection
$(2): \quad \forall k: k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
$(3): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

This definition also applies to modules, and also to vector spaces.

## Linguistic Note

The word epimorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix epi- meaning onto.

Thus epimorphism means onto (similar) structure.