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We study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as 'Rees criteria'. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.doi:10.1017/s0305004111000144 fatcat:mc4ulbuucrae5bu4c2wiaqn6oa