Codimension one cycles modulo rational equivalence form the classical group of divisors.
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Particularly important is the so-called " rational equivalence ".
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All cycles modulo rational equivalence form the Chow ring.
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Rational equivalence accomplishes the needs sketched above.
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They can be derived from the Hasse Minkowski theorem on the rational equivalence of quadratic forms.
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For the purposes of intersection theory, " rational equivalence " is the most important one.
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Cycles up to rational equivalence form a graded ring, the Chow ring, whose multiplication is given by the intersection product.
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The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence.
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