Definition:Epimorphism (Abstract Algebra)

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

Group Epimorphism
If both $\left({G, \circ}\right)$ and $\left({H, *}\right)$ are groups, then an epimorphism $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ is called a group epimorphism.

Ring Epimorphism
If both $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ are rings, then an epimorphism $\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$ is called a ring epimorphism.

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 iff:


 * $(1): \quad \phi$ 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.

Category Theory
In a category $\mathcal C$, an epimorphism is a morphism $\alpha \in \operatorname{mor}\mathcal C$ such that $\beta\alpha = \gamma \alpha$ implies $\beta = \gamma$ for all morphisms $\beta,\gamma \in \operatorname{mor}\mathcal C$ for which the composition is defined.

Group definition

 * : $\S 7.1$
 * : Chapter $\text{II}$
 * : $\S 47 \ \text{(b)}$

Ring definition

 * : $\S 5.24$
 * : $\S 2.2$: Definition $2.4$
 * : $\S 57$ Remarks: $\text{(a)} \ (2)$

R-Algebraic Structure definition

 * : $\S 28$