Definition:Endomorphism

Definition
An endomorphism is a homomorphism from an algebraic structure into itself.

Group Endomorphism
Let $$\left({G, \circ}\right)$$ be a group.

Then an endomorphism $$\phi: \left({G, \circ}\right) \to \left({G, \circ}\right)$$ is called a group endomorphism.

Ring Endomorphism
Let $$\left({R, +, \circ}\right)$$ be a ring.

Then an endomorphism $$\phi: \left({R, +, \circ}\right) \to \left({R, +, \circ}\right)$$ is called a ring endomorphism.

R-Algebraic Structure Endomorphism
Let $$\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$$ be an $R$-algebraic structure.

Then an endomorphism $$\phi: \left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R \to \left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$$ is called an $$R$$-algebraic structure endomorphism.

Group definition

 * : $$\S 7.1$$
 * : $$\S 1.2$$: Ring Example $$10$$

Ring definition

 * : $$\S 2.2$$: Definition $$2.4$$

R-Algebraic Structure definition

 * : $$\S 28$$