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

 * : $\S 28$