Definition:Homomorphism (Abstract Algebra)

Definition
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.

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

F-Homomorphism
Let $R$, $R^\prime$ be rings with unity.

Let $F$ be a subfield of both $R$ and $R^\prime$.

Then a ring homomorphism $\varphi: R \to R^\prime$ is called an $F$-homomorphism if:
 * $\forall a \in F: \varphi\left({a}\right) = a$.

That is, $\varphi \restriction_F = 1_F$ where:
 * $\varphi \restriction_F$ is the restriction of $\varphi$ to $F$;
 * $1_F$ is the identity mapping on $F$.

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

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) $\forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$.

Note that this definition also applies to modules and vector spaces.

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


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

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

We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as:
 * $\forall \left({x, y}\right) \in S \times S: \left({\phi \times \phi}\right) \left({x, y}\right) = \left({\phi \left({x}\right), \phi \left({y}\right)}\right)$

Hence we can state that $\phi$ is a homomorphism iff:
 * $\ast \left({\left({\phi \times \phi}\right) \left({x, y}\right)}\right) = \phi \left({\circ \left({x, y}\right)}\right)$

using the notation $\circ \left({x, y}\right)$ to denote the operation $x \circ y$.

The point of doing this is so we can illustrate what is going on in a commutative diagram:


 * Homomorphism.png

Thus we see that $\phi$ is a homomorphism iff both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.

Also see

 * Epimorphism: a surjective homomorphism;


 * Monomorphism: an injective homomorphism;


 * Isomorphism: a bijective homomorphism;


 * Endomorphism: a homomorphism from an algebraic structure to itself;


 * Automorphism: an isomorphism from an algebraic structure to itself.

Group definition

 * : $\S 7.1$
 * : $\S 1.10$
 * : Chapter $\text{II}$
 * : $\S 47$
 * : $\S 8$: Definition $8.1$

Ring definition

 * : $\S 2.2$: Definition $2.4$
 * : $\S 57$

R-Algebraic Structure definition

 * : $\S 28$