# Definition:Epimorphism (Abstract Algebra)

## Definition

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

### Semigroup Epimorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ 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 $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ 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 $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ 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
$(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
$(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$

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.