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Definition: The analog for a ring R of a vector space over a field is called a module. Let R be a ring. An R-module V is an abelian group with a law of composition written +, and a scalar multiplication RxV -> V, written r,v -> rv, that satisfy these axioms: 1v=v, (rs)v=r(sv), (r+s)v=rv+sv, and r(v+v')=rv+rv', for all r and s in R and all v and v' in V.
Source: Algebra, second edition by Michael Artin