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) $$\phi$$ is a surjection;
 * 2) $$\forall k: k \in \left[{1 \, . \, . \, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$$;
 * 3) $$\forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$$.

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

Group definition

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

Ring definition

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

R-Algebraic Structure definition

 * : $$\S 28$$