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Balanced module
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<h2 id="examples-and-properties">Examples and properties</h2> Examples <ul><li><a href="/facts/Semisimple_ring/9KASYk35">Semisimple rings</a> are balanced.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li>Every nonzero right <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> over a <a href="/facts/Simple_ring/5ndCExsb">simple ring</a> is balanced.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li>Every <a href="/facts/Faithful_module/aTaHR5W7">faithful module</a> over a <a href="/facts/Quasi-Frobenius_ring/noMdchsX">quasi-Frobenius ring</a> is balanced.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>The <a href="/facts/Double_centralizer_theorem/q5Lo1mgP">double centralizer theorem</a> for right Artinian rings states that any <a href="/facts/Simple_module/00roKQrG">simple</a> right <i>R</i> module is balanced.</li> <li>The paper (Dlab & Ringel 1972) contains numerous constructions of nonbalanced modules.</li> <li>It was established in (Nesbitt & Thrall 1946) that <a href="/facts/Uniserial_ring/hB2LGmBG">uniserial rings</a> are balanced. Conversely, a balanced ring which is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> as a module over its <a href="/facts/Center_of_a_ring/kJcejgkj">center</a> is uniserial.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>Among commutative Artinian rings, the balanced rings are exactly the <a href="/facts/Quasi-Frobenius_ring/noMdchsX">quasi-Frobenius rings</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul> Properties <ul><li>Being "balanced" is a categorical property for modules, that is, it is preserved by <a href="/facts/Morita_equivalence/vEVRJSJw">Morita equivalence</a>. Explicitly, if <i>F</i>(–) is a Morita equivalence from the category of <i>R</i> modules to the category of <i>S</i> modules, and if <i>M</i> is balanced, then <i>F</i>(<i>M</i>) is balanced.</li> <li>The structure of balanced rings is also completely determined in (Dlab & Ringel 1972), and is outlined in (Faith 1999, pp. 222–224).</li> <li>In view of the last point, the property of being a balanced ring is a Morita invariant property.</li> <li>The question of which rings have all finitely generated right <i>R</i> modules balanced has already been answered. This condition turns out to be equivalent to the ring <i>R</i> being balanced.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Camillo, Victor P. (1970), "Balanced rings and a problem of Thrall", <i>Trans. Amer. Math. Soc.</i>, 149: 143–153, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fs0002-9947-1970-0260794-0">10.1090/s0002-9947-1970-0260794-0</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0260794">0260794</a></li></ul> <ul><li>Bourbaki, Nicolas (1973), <i>Algébre, Ch. 8: Modules et Anneaux Semi-Simples</i>, p. 50, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-2-7056-1261-0</li></ul> <ul><li>Camillo, V. P.; Fuller, K. R. (1972), "Balanced and QF-1 algebras", <i>Proc. Amer. Math. Soc.</i>, 34 (2): 373–378, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fs0002-9939-1972-0306256-0">10.1090/s0002-9939-1972-0306256-0</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9939">0002-9939</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0306256">0306256</a></li></ul> <ul><li>Cunningham, R. S.; Rutter, E. A. Jr. (1972), "The double centralizer property is categorical", <i>Rocky Mountain J. Math.</i>, 2 (4): 627–629, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1216%2Frmj-1972-2-4-627">10.1216/rmj-1972-2-4-627</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0035-7596">0035-7596</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0310017">0310017</a></li></ul> <ul><li>Dlab, Vlastimil; Ringel, Claus Michael (1972), <a href="https://pub.uni-bielefeld.de/record/1775744">"Rings with the double centralizer property"</a>, <i>J. Algebra</i>, 22 (3): 480–501, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2872%2990163-9">10.1016/0021-8693(72)90163-9</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0306258">0306258</a></li></ul> <ul><li>Faith, Carl (1999), <i>Rings and things and a fine array of twentieth century associative algebra</i>, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0993-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1657671">1657671</a></li> <li>Lam, T.Y. (2001), <i>A first course in noncommutative rings</i>, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4419-8616-0">10.1007/978-1-4419-8616-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95183-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1838439">1838439</a></li> <li>Nesbitt, C. J.; Thrall, R. M. (1946), "Some ring theorems with applications to modular representations", <i>Ann. of Math.</i>, 2, 47 (3): 551–567, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969092">10.2307/1969092</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969092">1969092</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0016760">0016760</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>The definitions of balanced rings and modules appear in (Camillo 1970), (Cunningham & Rutter 1972), (Dlab & Ringel 1972), and (Faith 1999). - Camillo, Victor P. (1970), "Balanced rings and a problem of Thrall", Trans. Amer. Math. Soc., 149: 143–153, doi:10.1090/s0002-9947-1970-0260794-0, ISSN 0002-9947, MR 0260794 <a href="https://doi.org/10.1090%2Fs0002-9947-1970-0260794-0" target="_blank">https://doi.org/10.1090%2Fs0002-9947-1970-0260794-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bourbaki 1973, §5, No. 4, Corrolaire 2. - Bourbaki, Nicolas (1973), Algébre, Ch. 8: Modules et Anneaux Semi-Simples, p. 50, ISBN 978-2-7056-1261-0 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lam 2001, p.37. - Lam, T.Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439 <a href="https://doi.org/10.1007%2F978-1-4419-8616-0" target="_blank">https://doi.org/10.1007%2F978-1-4419-8616-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Camillo & Fuller 1972. - Camillo, V. P.; Fuller, K. R. (1972), "Balanced and QF-1 algebras", Proc. Amer. Math. Soc., 34 (2): 373–378, doi:10.1090/s0002-9939-1972-0306256-0, ISSN 0002-9939, MR 0306256 <a href="https://doi.org/10.1090%2Fs0002-9939-1972-0306256-0" target="_blank">https://doi.org/10.1090%2Fs0002-9939-1972-0306256-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Faith 1999, p.223. - Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, ISBN 0-8218-0993-8, MR 1657671 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1657671" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1657671</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Camillo 1970, Theorem 21. - Camillo, Victor P. (1970), "Balanced rings and a problem of Thrall", Trans. Amer. Math. Soc., 149: 143–153, doi:10.1090/s0002-9947-1970-0260794-0, ISSN 0002-9947, MR 0260794 <a href="https://doi.org/10.1090%2Fs0002-9947-1970-0260794-0" target="_blank">https://doi.org/10.1090%2Fs0002-9947-1970-0260794-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Dlab & Ringel 1972. - Dlab, Vlastimil; Ringel, Claus Michael (1972), "Rings with the double centralizer property", J. Algebra, 22 (3): 480–501, doi:10.1016/0021-8693(72)90163-9, ISSN 0021-8693, MR 0306258 <a href="https://pub.uni-bielefeld.de/record/1775744" target="_blank">https://pub.uni-bielefeld.de/record/1775744</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>