A free module is a module that has a basis, meaning it is isomorphic to a direct sum of copies of its ring, much like a vector space is to its field. This property allows for the representation of any element in the module as a unique linear combination of basis elements, which establishes the module's structure and facilitates computations. Free modules play a significant role in the study of linear algebra and homological algebra, particularly in relation to Ext and Tor functors.
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