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Category of modules
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<h2 id="properties">Properties</h2> <p>The categories of left and right modules are <a href="/facts/Abelian_category/4WN473Yc">abelian categories</a>. These categories have <a href="/facts/Enough_projectives/sKWPFB6u">enough projectives</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and <a href="/facts/Enough_injectives/JGABzfyw">enough injectives</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <a href="/facts/Mitchell%2527s_embedding_theorem/6Mmx0k9e">Mitchell's embedding theorem</a> states every abelian category arises as a <a href="/facts/Full_subcategory/k4lEeDDb">full subcategory</a> of the category of modules over some ring. </p><p><a href="/facts/Projective_limit/f6QecRUu">Projective limits</a> and <a href="/facts/Inductive_limit/V3JuHnYK">inductive limits</a> exist in the categories of left and right modules.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>, together with the <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product of modules</a> ⊗, the category of modules is a <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric monoidal category</a>. </p> <h2 id="objects">Objects</h2> <p>A <a href="/facts/Monoid_object/Dt2JUwRH">monoid object</a> of the category of modules over a commutative ring <i>R</i> is exactly an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over <i>R</i>. </p><p>A <a href="/facts/Compact_object_(mathematics)/npMr7u6t">compact object</a> in <i>R</i>-Mod is exactly a finitely presented module. </p> <h2 id="category-of-vector-spaces">Category of vector spaces</h2> <p class="note">See also: <a href="/facts/FinVect/24D2dmvG">FinVect</a></p> <p>The <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> <i>K</i>-Vect (some authors use Vect<i>K</i>) has all <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>K</i> as objects, and <a href="/facts/Linear_map/Q1PBKNVV"><i>K</i>-linear maps</a> as morphisms. Since vector spaces over <i>K</i> (as a field) are the same thing as <a href="/facts/Module_(algebra)/jP0LIhFL">modules</a> over the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>K</i>, <i>K</i>-Vect is a special case of <i>R</i>-Mod (some authors use Mod<i>R</i>), the category of left <i>R</i>-modules. </p><p>Much of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> concerns the description of <i>K</i>-Vect. For example, the <a href="/facts/Dimension_theorem_for_vector_spaces/1dQJF9fT">dimension theorem for vector spaces</a> says that the <a href="/facts/Isomorphism_class/wDUgsJQ8">isomorphism classes</a> in <i>K</i>-Vect correspond exactly to the <a href="/facts/Cardinal_number/ccoKwU4R">cardinal numbers</a>, and that <i>K</i>-Vect is <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalent</a> to the <a href="/facts/Subcategory/k4lEeDDb">subcategory</a> of <i>K</i>-Vect which has as its objects the vector spaces <i>K</i><i>n</i>, where <i>n</i> is any cardinal number. </p> <h2 id="generalizations">Generalizations</h2> <p>The category of <a href="/facts/Sheaves_of_modules/grSMBXho">sheaves of modules</a> over a <a href="/facts/Ringed_space/JyeaTMtX">ringed space</a> also has enough injectives (though not always enough projectives). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Algebraic_K-theory/0RY5M49W">Algebraic K-theory</a> (the important invariant of the category of modules.)</li> <li><a href="/facts/Category_of_rings/pIwY32bc">Category of rings</a></li> <li><a href="/facts/Derived_category/pBQHnFYo">Derived category</a></li> <li><a href="/facts/Module_spectrum/569pLk5l">Module spectrum</a></li> <li><a href="/facts/Category_of_graded_vector_spaces/Ff1maywV">Category of graded vector spaces</a></li> <li><a href="/facts/Category_of_representations/Lxr3DLnt">Category of representations</a></li> <li><a href="/facts/Change_of_rings/RYGsTsYk">Change of rings</a></li> <li><a href="/facts/Morita_equivalence/vEVRJSJw">Morita equivalence</a></li> <li><a href="/facts/Stable_module_category/slNEu8ql">Stable module category</a></li> <li><a href="/facts/Eilenberg%25E2%2580%2593Watts_theorem/2h8I7ILQ">Eilenberg–Watts theorem</a></li></ul> <h3>Bibliography</h3> <ul><li><a href="/facts/Bourbaki_group/qRJhs9nr">Bourbaki</a>. "Algèbre linéaire". <i>Algèbre</i>.</li> <li>Dummit, David; Foote, Richard. <i>Abstract Algebra</i>.</li> <li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (September 1998). <i>Categories for the Working Mathematician</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 5 (second ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98403-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0906.18001">0906.18001</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://ncatlab.org/nlab/show/Mod">Mod</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"module category in nLab". ncatlab.org. <a href="http://ncatlab.org/nlab/show/module+category" target="_blank">http://ncatlab.org/nlab/show/module+category</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>trivially since any module is a quotient of a free module. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Dummit & Foote, Ch. 10, Theorem 38. - Dummit, David; Foote, Richard. Abstract Algebra. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bourbaki, § 6. - Bourbaki. "Algèbre linéaire". Algèbre. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>