Let $R$ be a commutative and local ring, and $M$ is a right $R$-module and $N$ is a left $R$-module. If $M$ and $N$ are finitely generated, then $M\otimes_{R} N=0$ if and only if $M=0$ or $N=0$.   If $R$ is not local, then this proposition is true? Why?  I want to prove this proposition. But I don't know how to start. Please help me.   Thank you.
If $R$ is a local ring and $M$ and $N$ are finitely generated $R$-modules such that $M\otimes N=0$ then how does it follow from Nakayama's lemma that either $M=0$ or $N=0$?    This is an exercise in Atiyah and Macdonald. The part I could not show in the hints is $({M{\otimes}_R N)}_{k}=0$ implies $M_{k}{\otimes }_{k} N_{k}=0$, where $k=R/\mathfrak m$ and $\mathfrak m$ is the maximal ideal of $R$.
The cardinality of $\{f\subseteq\mathbb R^2\mid f\colon\mathbb{R\to R}\text{ is a function}\}$ is strictly greater than the cardinality of $\mathbb R$ make tail -f exit on a broken pipe I want to watch a file until a bit of text appears  I found this answer:   but when I tried it on ubuntu, it doesn't exit:  $ echo one &gt; test.txt $ echo two &gt;&gt; test.txt $ echo three &gt;&gt; test.txt $ echo four &gt;&gt; test.txt $ cat test.txt | sed '/w/ q' one two $   works as expected.  however when I try to tail the file  $ tail -f test.txt | sed '/w/ q' one two   it never exits.  tail does not stop even though the pipe is broken.  Does anybody know how to make tail exit when sed exits? A google search gives number of methods to hide ALL tabs in chrome , but I want to hide 1 particular tab / change the display name of that tab.   Reason: I have an online music playing website open, which keeps showing tracks in the tab name and I just want to remove it from view of my work. Could "them" mean "those"? Conveying "under the radar" in a less colloquial way Determine if the convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ is uniform. $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ converges pointwise in a set $E = [0, \infty)$ to $f(x) = x$.   This problem reminds me a lot of how $\frac{x}{n}$ fails to converge uniformly to $f=0$ on $[0,\infty]$, but does converge uniformly on $[0,a]$. However, when we proved that we used the definition of uniform convergence. I do not wish to do that here.  I wish to use the Weierstrass test here. And that is where I'm stuck. I cannot find $M_n = \sup\limits_{x \in E} |f_n(x)-f(x)|$. Can someone help me out here? I wanted to apply texture on some house following an tutorial, I'm sure I did all what was needed but still don't see the texture.  Here's a screenshot: Display data from geoserver on click map with openlayers I tried to display data with popup from Geoserver or Postgres to my map, but it dosen't work at all. Can anyone please help me? I used this code:      function Info(e)          {                 if (e.features &amp;&amp; e.features.length)                  {                      //console.log(e.features[0].attributes);                     var popupId = e.xy.x + "," + e.xy.y;                     var popup = popups[popupId];                     if (!popup || !popup.map) {                     popup = new OpenLayers.Popup.FramedCloud(                     popupId,                      map.getLonLatFromPixel(e.xy),                     null,                     " ",                     null,                     true,                     function(e) {                         delete popups[this.id];                         this.hide();                         OpenLayers.Event.stop(e);                     }                 );                 popups[popupId] = popup;                 map.addPopup(popup, true);                 }                popup.setContentHTML(popup.contentHTML + "nom_user:" +e.features[0].attributes.nomservice +" region_user :"+ e.features[0].attributes.region);             popup.show();                                    }       selectcltr = {         click: new OpenLayers.Control.WMSGetFeatureInfo({             url: 'http://localhost/geoserver/wms',              title: 'Identify features by clicking',             layers: [ser],             queryVisible: true         }),         hover: new OpenLayers.Control.WMSGetFeatureInfo({             url: 'http://localhost/geoserver/wms',              title: 'Identify features by clicking',             layers: [ser],             hover: true,             // defining a custom format options here             formatOptions: {                 typeName: 'projet',                  featureNS: 'http://www.openplans.org/topp'             },             queryVisible: true         })     } What happens to water level when ice XII melts? There exists this famous idea that if all floating icebergs melt, water level will stay the same (because the water replaced by ice is the volume of the melted ice). Now,   Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ? Is this always so on/in earth ?   EDIT:  I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. "Ice" is always floating or sunken(if possible).  To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)?  EDIT 2:  Wikipedia: "Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)".   And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the "lake" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen? Why "grand theft auto", not "grand auto theft"? There is a video game series called "Grand Theft Auto".  According to its :     The name of the series references the term used in the US for motor   vehicle theft.      [...]      Motor vehicle theft or, grand theft auto, is the criminal act of   stealing or attempting to steal a car (or any other motor vehicle).   I always wondered why this term sounds so strange.   Shouldn't it be "grand auto theft" or maybe "grand theft of auto" instead? Are there any other terms with strange words order and/or missing prepositions like this? How do I get a platform-dependent new line character? How do I get a platform-dependent newline in Java? I can’t use "\n" everywhere. Zero divisor in $R[x]$ Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=ba_n=0$? How to make terminal transparent while still having its own background image? Is there a way to give terminal specific background image and make it 50% transparent at the same time? Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$? Reading about photons I hear different explanations like "elementary particle", "probability cloud", "energy quanta" and so forth. Since probably no one has ever seen a photon (if "seen" it supposedly - and rather conveniently - ceases to exist) but many experiments seem to verify its properties (or are they maybe adjusted to fit the experiment). I thus can't help wondering if the "photon" is then just a physical/mathematical tool with inexplicable properties (like zero mass - but affected by gravity fields - and constant speed c in space) invented to explain some otherwise unexplainable phenomem and to supplement the elementary particles and their interactions. In short: Are they real or imaginary!  Does anybody know? Or maybe the answer is "blowing in the wind" because to most physicist it probably just does'nt matter as long as it works (as the alternative healers say). Sorry if I seem a little sarcastic here and there. After a botched upgrade from Ubuntu 14.04 to Ubuntu 16.04 I had to wipe my system and start all over. This time I decided to go with a much less CPU intensive desktop environment and installed Lubuntu 16.04-1.  My problem: when I run Steam from the menu shortcut it acts like it is doing nothing. However it does show up in the task manager. If I just type steam in the terminal I get the following.  michael@michael-HP-Pavilion-dm1-Notebook-PC:~$ steam Running Steam on ubuntu 16.04 64-bit STEAM_RUNTIME is enabled automatically grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc /bin/bash: /home/michael/.local/share/Steam/ubuntu12_32/steam-runtime/amd64/lib/x86_64-linux-gnu/libtinfo.so.5: no version information available (required by /bin/bash) grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc awk: /home/michael/.local/share/Steam/ubuntu12_32/steam-runtime/amd64/lib/x86_64-linux-gnu/libtinfo.so.5: no version information available (required by /lib/x86_64-linux-gnu/libreadline.so.6) Installing breakpad exception handler for appid(steam)/version(1468023329) libGL error: unable to load driver: r600_dri.so libGL error: driver pointer missing libGL error: failed to load driver: r600 libGL error: unable to load driver: swrast_dri.so libGL error: failed to load driver: swrast   Then nothing happens including steam doesn't show up in the task manager. Now I did find the command.  LD_PRELOAD='/usr/$LIB/libstdc++.so.6' DISPLAY=:0 steam   This does run Steam but it is not a real solution in my eyes. How can I fix it? Index of the center of a group in the group is not a prime number The question is to prove index of $Z(G)$ in $G$ is not a prime number.    We know that $|G:Z(G)|=|G:C_G (x)||C_G (x):Z(G)|$ where $C_G (x)$ means centralizer of $x \in G$  I want to mention that we do not want to use $G⁄Z(G)$ theorem.   We may assume that $|G:Z(G)|=p$ where $p$ is a prime. Redirect based on country Let's say I have a site that has multiple versions for different countries, e.g.:  www.example.com/us www.example.com/uk www.example.com/ca   and I want to use IP geolocation to redirect users who go to www.example.com to the appropriate country version of the site.  What is the best way to do that without confusing crawlers such as Googlebot or running into other SEO type trouble? How to increase swap space? I have, by default, 250MB of swap space in Ubuntu, and I want to expand it to a larger size. I need 800MB, which I think will be enough to open several applications without having to hit the current limit of swap. I hope someone can help me. Specify CSS class names on SVG paths using Illustrator Is there a way to edit SVG files in Illustrator where you can specify a CSS class for each path?  I already know in Illustrator that if you give the layer name an actual name, Illustrator will use that name as the path's ID, which is fine if you don't plan on reusing the icon more than once on the same page.  My current workaround is to just use the IDs method, but then later convert the IDs to classes in my code editor, but it's pretty annoying to have to do every time I generate my SVG sprite.  If that's currently not possible in Illustrator, are there any other apps that will allow you to specify that? Or maybe a Grunt or Gulp package?  It looks like you might be able to do it with Inkscape with a , so I might look at that if there's no other good solutions out there. How can I repeat the header but not the caption with longtable? I am trying to set up a longtable where the header is repeated on every new page but the caption only appears once at the top of the page.  The table is defined as follows:  \begin{longtable}{|l|l|l|} \caption{the table caption} \label{tab:11swpkg}\\ \hline \rowcolor{gray} HeaderCol1 &amp; HeaderCol2 &amp; HeaderCol3\endhead     \hline\hline Item11 &amp; Item12 &amp; Item13 \\ \hline Item21 &amp; Item22 &amp; Item23 \\ \hline Item31 &amp; Item32 &amp; Item33 \\ \hline \end{longtable}   This nearly works because the header is indeed repeated after a page break but so is the caption. How can I get rid of the caption after a page break?