Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Countably generated module
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over a (not necessarily <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a>) <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> is countably generated if it is <a href="/facts/Generating_set_of_a_module/mgvpF9Yo">generated</a> as a module by a <a href="/facts/Countable/Wj4DmWPv">countable</a> subset. The importance of the notion comes from <a href="/facts/Kaplansky%2527s_theorem_on_projective_modules/LhRhNVbg">Kaplansky's theorem</a> (Kaplansky 1958), which states that a <a href="/facts/Projective_module/lHymqhNC">projective module</a> is a <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">direct sum</a> of countably generated modules. </p><p>More generally, a module over a possibly <a href="/facts/Non-commutative_ring/3jmll7q7">non-commutative</a> ring is projective <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> (i) it is <a href="/facts/Flat_module/1WFrnJP3">flat</a>, (ii) it is a direct sum of countably generated modules and (iii) it is a Mittag-Leffler module. (Bazzoni–Stovicek) </p> <ul><li><a href="/facts/Irving_Kaplansky/nRLyp7K9">Kaplansky, Irving</a> (1958). "Projective Modules". <i>Annals of Mathematics</i>. 68 (2): 372–377. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970252">10.2307/1970252</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F101124">10338.dmlcz/101124</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970252">1970252</a>.</li> <li>Bazzoni, Silvana; Šťovíček, Jan (2012). <a href="https://doi.org/10.1090%2FS0002-9939-2011-11070-0">"Flat Mittag-Leffler modules over countable rings"</a>. <i><a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a></i>. 140 (5): 1527–1533. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1007.4977">1007.4977</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-2011-11070-0">10.1090/S0002-9939-2011-11070-0</a>.</li></ul>