 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.The general notion of a congruence relation can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.In this setting, a congruence relation is an equivalence relation formula_32 on an algebraic structure that satisfies
for every formula_1-ary operation formula_45 and all elements formula_46 such that formula_47 for each formula_48

If formula_49 is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation formula_50 defined by

is a congruence relation.