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

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

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