Is there an example of a commutative ring $ A $ with unit $ 1_A $ and an $A$-module map : $ f : A^{n+1} \to A^n $ such that $f$ is injective ?  If the answer is yes, when is this enunciation false ?  Thanks in advance for your help.
I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold.    This is exercise 2.11 from Atiyah and MacDonald's Introduction to Commutative Algebra, but unfortunately all the solutions available online are either very sparse with regard to this question or seem to use (to my surprise) a generalized version of Cramer's rule. Is there perhaps some other "cleaner" approach to solving this problem?
I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers resolv.conf not working properly with Ethernet in Ubuntu I have a Lenovo X200 laptop on which I am running Ubuntu 9.10. 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I've run the One-Sample Kolmogorov-Smirnov test to check if the distribution is either exponential or poisson, but I got negative results.  Is it possible that the One-Sample Kolmogorov-Smirnov test, it's not the correct test to work with this count data.  What is the best way to identify a distribution for this kind of data? i have a tiny little problem with xslt, js and html entities, eg. within a template:  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &amp;lt; 5; i++) {         //            ^^^ js error     } &lt;/script&gt;  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &lt; 5; i++) {         //            ^ xslt error     } &lt;/script&gt;  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     // &lt;![CDATA[     for (var i = 0; i &lt; 5; i++) {         //            ^ becomes &amp;lt;     }     // ]]&gt; &lt;/script&gt;   &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &lt;xsl:value-of disable-output-escaping="yes" select="string('&amp;lt;')"/&gt; 5; i++) {         // works of course     } &lt;/script&gt;   does anyone have an idea where my problem could come from? i always thought the xslt processor would leave the content of a &lt;script/&gt; element unescaped when using the html output method ...  i run libxslt2 version 1.1.24 on OSX which was installed using macportsports ... 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I downloaded 12.04LTS LiveCD yesterday and made a CD and a bootable USB stick.  With both USB and LiveCD, occasionally I can boot into 12.04.  But many times, the boot process stuck at some point.  For USB boot, sometime it works.  But other times, it seems to stuck right after seeing the message:  NMI watchdog enabled. takes one hw-pmu counted [0.408590] #2   The number in [] changes.  Some times it stuck at #2, and sometimes at #3.  For LiveCD boot, sometime it works.  But when it stuck, it stuck at different places.  At times, the boot process go directly to a blank screen with a blink '_' at up left corner.  Sometime it went a bit further, after showing options to try/install ubuntu then stuck with a blank screen with blink '_'.  Anything I can do to get around of this? Can I install Kali tools on Ubuntu? How do I change a switched outlet to a half-hot outlet? I am trying to change a switched outlet to a half-hot outlet. 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