About injective hull - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:10:26Z http://mathoverflow.net/feeds/question/54065 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54065/about-injective-hull About injective hull ashpool 2011-02-02T05:22:25Z 2012-12-01T21:38:28Z <p>Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?</p> http://mathoverflow.net/questions/54065/about-injective-hull/54066#54066 Answer by Karl Schwede for About injective hull Karl Schwede 2011-02-02T05:27:55Z 2011-02-02T05:27:55Z <p>Yes, take $A = k[[x]]$ and $M = A/(x)$. Then as a $k = A/(x) = A/\text{Ann}(M)$-module, the injective hull of $k$ is $k$. As an $A$-module, the injective hull is much much bigger.</p> http://mathoverflow.net/questions/54065/about-injective-hull/54200#54200 Answer by Chris Leary for About injective hull Chris Leary 2011-02-03T14:44:18Z 2011-02-03T14:44:18Z <p>I'll follow up on what Karl said with an example closer to my own experience. Let Z be the ring of integers and p a positive prime. Then Z/pZ is injective as a Z/pZ - module, being a vector space over a field, whence Z/pZ is its own injective envelope (hull) as a Z/pZ module. However, the injective envelope of Z/pZ as an abelian group is Z(p^{infty}), which gives witness to Karl's statement that the injective envelope over A can be much larger than the injective envelope over A/ann(M). You can play this game with A any commutative Noetherian ring with 1, ann(M) = any maximal ideal of R, and M = A/I where I is the chosen maximal ideal. Karl's example presents very limited choice for I since k[[x]] is local. I think Proposition 2.27 and Lemma 4.24 of "Injective Modules" by Sharpe and Vamos present enough to figure out what is going on in the general case.</p> http://mathoverflow.net/questions/54065/about-injective-hull/115117#115117 Answer by razi for About injective hull razi 2012-12-01T21:38:28Z 2012-12-01T21:38:28Z <p>whate\ injective hull of field K is K?</p>