Direct sum of injective modules over non-Noetherian rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:41:22Z http://mathoverflow.net/feeds/question/53928 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53928/direct-sum-of-injective-modules-over-non-noetherian-rings Direct sum of injective modules over non-Noetherian rings Aaron Bennet 2011-01-31T22:55:07Z 2012-02-17T05:59:41Z <p>Hi. I know, by the Bass-Papp theorem, that if every direct sum of injective \$R\$-modules is injective then \$R\$ is Noetherian. I would like to know if there exists a direct sum of injective \$R\$-modules injective with \$R\$ non-Noetherian. Of course if the sum is a finite sum of injective modules, then it is injective; so I assume that the sum is infinite (let's say that all the modules are nonzero).</p> http://mathoverflow.net/questions/53928/direct-sum-of-injective-modules-over-non-noetherian-rings/53935#53935 Answer by Greg Marks for Direct sum of injective modules over non-Noetherian rings Greg Marks 2011-02-01T00:44:43Z 2011-02-01T00:44:43Z <p>A module is called <i>\$\Sigma\$-injective</i> if a direct sum of arbitrarily (equivalently, countably infinitely) many copies of that module is injective.&#160; So it suffices to find an example of a \$\Sigma\$-injective module over a non-noetherian ring.&#160; Apart from silly examples such as a direct product of two rings one of which is one-sided noetherian and the other of which is not, the main theorem of C. Megibben, &#8220;Countable injective modules are \$\Sigma\$-injective,&#8221; <i>Proc. Amer. Math. Soc.</i> <b>84</b> (1982), no. 1, 8&#8211;10, says what the title indicates.&#160; This gives all sorts of examples of \$\Sigma\$-injective modules over non-noetherian rings.&#160;</p>
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