Definition:Homomorphism (Abstract Algebra)

Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a mapping from one algebraic structure $$\left({S, \circ}\right)$$ to another $$\left({T, *}\right)$$.

If $$\circ$$ has the morphism property under $$\phi$$, that is:

$$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$$

then $$\phi$$ is a homomorphism.

This can be generalised to algebraic structures with more than one operation:

Let: be algebraic structures.
 * $$\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$$
 * $$\left({T, *_1, *_2, \ldots, *_n}\right)$$

Let $$\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$$ be a mapping from $$\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$$ to $$\left({T, *_1, *_2, \ldots, *_n}\right)$$.

If, $$\forall k \in \left[{1 \,. \, . \, n}\right]$$, $$\circ_k$$ has the morphism property under $$\phi$$, that is:

$$\forall x, y \in S: \phi \left({x \circ_k y}\right) = \phi \left({x}\right) *_k \phi \left({y}\right)$$

then $$\phi$$ is a homomorphism.

Group Homomorphism
If both $$\left({S, \circ}\right)$$ and $$\left({T, *}\right)$$ are groups, then a homomorphism $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ is called a group homomorphism.

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

F-Homomorphism
Let $$R$$, $$R^'$$ be rings, both containing the field $$F$$. Then a ring homomorphism $$\varphi : R\rightarrow R'$$ is called an $$F$$-homomorphism if $$\varphi(a)=a~\forall ~a\in F$$; that is to say $$\varphi \restriction_F = 1_F$$.

Furthermore, if $$\varphi$$ is an isomorphism, we call it an $$F$$-isomorphism and write $$R \cong_F R^'$$

R-Algebraic Structure Homomorphism
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 $$\phi$$ is an $$R$$-Algebraic Structure homomorphism iff:


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

Image
As a homomorphism is a mapping, and therefore a relation, we define the image of a homomorphism in the same way:

$$\mathrm{Im} \left({\phi}\right) = \left\{{t \in T: \exists s \in S: t = \phi \left({s}\right)}\right\}$$