Definition:Epimorphism (Abstract Algebra)

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This page is about Epimorphism in the context of Abstract Algebra. For other uses, see Epimorphism.

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.


Also see


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.


Sources