3. Factor algebras

For a group $G$ with normal subgroup $H$, we can form $G/H$. For a commutative ring $R$ with ideal $I$, we can form $R/I$. In general:

3.1 Definition. For an algebra ${\cal A} = \langle A; f _ 1,\dots, f _ m \rangle$ and $\theta \in$   Con$({\cal A})$, let ${\cal A}/\theta$ be the algebra whose elements are the blocks of $\theta$ and whose operations are defined as follows: For each basic operation $f _ i$ on A, define a corresponding operation $f _ i$ on ${\cal A}/\theta$ by

$f _ i (\bar a _ 1,\dots, \bar a _ {n _ i}) = \overline f _ i (a _ 1,\dots, a _ {n _ i})$.

This operation is well defined, since by the definition of a congruence relation the result does not depend on which representatives are chosen for the blocks. Just as for groups or rings, ${\cal A}/\theta$ is called a ``factor algebra'' or ``quotient algebra'' obtained by ``factoring out $\theta$''. Don't confuse this with the concept of a ``field of quotients''.

3.2 Definition. The natural map of ${\cal A}$ onto ${\cal A}/\theta$ is $\pi: {\cal A} \rightarrow {\cal A}/\theta$ given by $\pi(a) = \bar a$.

3.3 Proposition. The natural map of ${\cal A}$ onto ${\cal A}/\theta$ is a surjective homomorphism with kernel $\theta$.

This natural map can also be called the natural homomorphism or natural surjection. See Figure .

Figure: The natural homomorphism

3.4 Corollary. Every congruence relation is the kernel of some homomorphism.

3.5 Note. If $\theta _ 1 \subseteq \theta _ 2$, then there is a natural surjection ${\cal A}/\theta _ 1 \rightarrow {\cal A}/\theta _ 2$. To remember the direction of this map, think of ${\cal A}/\theta _ 1$ as bigger than ${\cal A}/\theta _ 2$, since in ${\cal A}/\theta _ 1$, less has been factored out.