Example: Rings

When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy
: and whenever and . For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

Relation with homomorphisms

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