Chow ring
Possible (and useful) adequate equivalence relations include "rational", "algebraic", "homological" and "numerical equivalence". They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical group of divisors. All cycles modulo rational equivalence form the Chow ring.