Definition:Endomorphism

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

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

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