Preservation of direct sums and finite generation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:55:14Z http://mathoverflow.net/feeds/question/81320 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81320/preservation-of-direct-sums-and-finite-generation Preservation of direct sums and finite generation Pierre-Yves Gaillard 2011-11-19T06:21:03Z 2011-11-19T06:21:03Z <p>I asked this <a href="http://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/" rel="nofollow">question</a> on Mathematics - Stack Exchange (MSE). Having figured out out how to handle the problem in an <strong>extremely</strong> particular case, I also posted it as an <a href="http://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/82958#82958" rel="nofollow">answer</a> (in the technical sense of the term) to my own question. Getting no other answer, I thought I could post the question on MathOverflow. For the reader's convenience, here is a copy and paste of the question. </p> <p>This is a follow up on this MSE <a href="http://math.stackexchange.com/questions/78161/hom-and-direct-sums" rel="nofollow">question</a>, asked by Evariste. </p> <p>Let $R$ be an associative ring with one. The word "module" shall mean <em>left</em> $R$-module. Say that a module $A$ <strong>preserves direct sums</strong> if the functor $\hom_R(A,?)$ does. </p> <p>The <strong>main question</strong> is</p> <blockquote> <p>Does the condition that $A$ preserves direct sums imply that $A$ is finitely generated? </p> </blockquote> <p>The converse is clear: see this MSE <a href="http://math.stackexchange.com/questions/78161/hom-and-direct-sums/78178#78178" rel="nofollow">answer</a>. </p> <p>As observed by Mariano Suárez-Alvarez in a comment to this MSE <a href="http://math.stackexchange.com/questions/78161/hom-and-direct-sums/78178#78178" rel="nofollow">answer</a>, if $A$ can be written as the union of an increasing sequence $(A_n)_{n\in\mathbb N}$ of submodules, then $A$ does <strong>not</strong> preserve direct sums. [The argument is described in the answer.]</p> <p>Say that $A$ is <strong>countably cofinal</strong> if it can be written as such a union. If $A$ is neither finitely generated nor countably cofinal, say that $A$ is <strong>uncountably cofinal</strong>. </p> <p>[Here is the motivation for this terminology. A group which can be written as the union of an increasing sequence of subgroups is called <em>countably cofinal</em>, and a group which is neither finitely generated nor countably cofinal, is called <em>uncountably cofinal</em>. Uncountably cofinal groups have been studied by Serre, Tits, MacPherson, Bergman, and many others: see this <a href="http://www.google.com/search?q=%22uncountable+cofinality%22&amp;hl=en&amp;safe=off#sclient=psy-ab&amp;hl=en&amp;safe=off&amp;source=hp&amp;q=%22uncountable+cofinality%22+serre+tits+macpherson+bergman&amp;pbx=1&amp;oq=%22uncountable+cofinality%22+serre+tits+macpherson+bergman&amp;aq=f&amp;aqi=&amp;aql=&amp;gs_sm=e&amp;gs_upl=51552l58613l2l59788l4l3l1l0l0l0l492l1142l2-1.0.2l4l0&amp;bav=on.2,or.r_gc.r_pw.r_cp.,cf.osb&amp;fp=5a3ef425d86707e8&amp;biw=1422&amp;bih=705" rel="nofollow">Google Search</a>. In particular, uncountably cofinal groups do exist.] </p> <p>The <strong>second question</strong> is: </p> <blockquote> <p>Do uncountably cofinal modules exist?</p> </blockquote>