Congruence relation

Remarks

Clause 2 is called the compatibility property. One often writes $a \mathrel{\theta} b$ or $a \equiv b \pmod{\theta}$ for $(a, b) \in \theta$. Every algebra has the trivial congruences $\Delta_A = \set{(a, a) : a \in A}$ and $\nabla_A = A \times A$.

Example: on the ring $(\Z, +, \cdot, -, 0, 1)$, fix $n \ge 1$ and let $a \mathrel{\theta} b$ when $n$ divides $a - b$. If $n$ divides $a - a'$ and $b - b'$, it divides $(a + b) - (a' + b')$, $-a - (-a')$ and $ab - a'b' = a(b - b') + (a - a')b'$.

Non-example: on the group $(\Z, +, -, 0)$, let $a \mathrel{\theta} b$ when $a$ and $b$ are both positive, both negative, or both $0$. This is an equivalence relation, but $1 \mathrel{\theta} 2$ while $1 + (-1) = 0$ and $2 + (-1) = 1$ are not related.

In a group, the congruences correspond to the normal subgroups via $\theta \mapsto \set{a : a \mathrel{\theta} e}$. In a ring, they correspond to the two-sided ideals via $\theta \mapsto \set{a : a \mathrel{\theta} 0}$.