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Finitely generated module
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a finitely generated module is a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> that has a <a href="/facts/Finite_set/za1newlP">finite</a> <a href="/facts/Generating_set_of_a_module/mgvpF9Yo">generating set</a>. A finitely generated module over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>R</i> may also be called a finite <i>R</i>-module, finite over <i>R</i>, or a module of finite type. </p><p>Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> the concepts of finitely generated, finitely presented and coherent modules coincide. </p><p>A finitely generated module over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> is simply a <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, and a finitely generated module over the <a href="/facts/Integer/PUwwrawx">integers</a> is simply a <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated abelian group</a>. </p>