(algebra, ring theory) An abelian group equipped with the operation of multiplication by an element of a ring (or another of certain algebraic objects), representing a generalisation of the concept of vector space with scalar multiplication.
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Modules over a ring are a generalization of abelian groups (which are modules over \textstyle\mathbb{Z}).
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1974, Thomas W. Hungerford, Algebra, Springer, page 168:
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Approximately forty-five years ago K. Morita presented the first major results on equivalences and dualities between categories of modules over a pair of rings.
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2004, Robert R. Colby, Kent R. Fuller, Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings), Cambridge University Press, page vii:
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One defines in like manner right K-modules and two-sided K-modules. If K is commutative, then every left K-module is automatically equipped with the structure of right and a two-sided K-module.
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2012, A. A. Kirillov, Elements of the Theory of Representations, Springer, page 29: