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Artin algebra
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<h2 id="dual-and-transpose">Dual and transpose</h2> <p>There are several different dualities taking finitely generated modules over Λ to modules over the <a href="/facts/Opposite_algebra/NY2dabJ7">opposite algebra</a> Λop. </p> <ul><li>If <i>M</i> is a left Λ-module then the right Λ-module <i>M</i>* is defined to be HomΛ(<i>M</i>,Λ).</li> <li>The dual <i>D</i>(<i>M</i>) of a left Λ-module <i>M</i> is the right Λ-module <i>D</i>(<i>M</i>) = Hom<i>R</i>(<i>M</i>,<i>J</i>), where <i>J</i> is the dualizing module of <i>R</i>, equal to the sum of the <a href="/facts/Injective_envelope/VoENIwTw">injective envelopes</a> of the non-isomorphic <a href="/facts/Simple_module/00roKQrG">simple</a> <i>R</i>-modules or equivalently the injective envelope of <i>R</i>/rad <i>R</i>. The dual of a left module over Λ does not depend on the choice of <i>R</i> (up to isomorphism).</li> <li>The transpose Tr(<i>M</i>) of a left Λ-module <i>M</i> is a right Λ-module defined to be the <a href="/facts/Cokernel/vYFQRPoh">cokernel</a> of the map <i>Q</i>* → <i>P</i>*, where <i>P</i> → <i>Q</i> → <i>M</i> → 0 is a minimal projective presentation of <i>M</i>.</li></ul> <ul><li>Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], <a href="https://books.google.com/books?isbn=0521599237"><i>Representation theory of Artin algebras</i></a>, Cambridge Studies in Advanced Mathematics, vol. 36, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-59923-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1314422">1314422</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0834.16001">0834.16001</a></li></ul>