Proving the Nilradical of a Commutative Ring is closed under addition.
if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
How to get the native version of Spotify running? Shutdown and User button missing in panel I faced with a problem after update,   I missed 2 buttons from above panel, System menu (including shutdown, suspend &amp; …) and User (including log out &amp; …)  how can I restore them !?... I'd like to find a place to store stuff while I'm out adventuring, without spending 5000 on a house (at least, until I can afford that...). Where can I safely store items without worrying about them disappearing?   There are two possible places that come to mind,  (to which I have the key and was told I could stay), and the small bedroom given to me at the  (which I was told was mine, despite seeing J'zaro sitting in it later). Can I store items safely in either of these places? I know, it's been asked before but I couldn't really get a satisfying answer for some details/backgrounds and I guess it's an important question for many.    Some application seem only to be running with Adobe Flash 15 (or at least I know it does, because I have that version on my Windows Partition).  The official Adobe website says:     NOTE: Adobe Flash Player 11.2 will be the last version to target Linux as a supported platform. Adobe will continue to provide security backports to Flash Player 11.2 for Linux.   Some search on the web like "Ubuntu Adobe Flash 15" or similar does not return any good websites.  I found something that says   (using a "pepper-based version"), and  .   Is it impossible to use this for Firefox too? Would it be a privacy risk?  Are there no alternative projects (I know it's closed source but there are so many so very bright people out there using something similar to wine but just for the plugin....)?  How about the Mac-Version? Shouldn't someone be able to port that one to Linux? Is there any SE Site where Questions related to Hacking are on-topic? Why would a slug thrower be more effective against a lightsaber/Jedi than a blaster? From  question about gunpowder in the Star Wars universe it is  that the shot from a  slug thrower is not completely deflectable by a lightsaber. This seems to be listed as an advantage (over blasters) against lightsabers and Jedi.     For example, during the days of the Galactic Republic, mercenaries fearing an intervention by Jedi Knights used rapid-fire slugthrowers that were impossible to completely deflect, unlike blaster bolts.   This doesn't make much sense though. A slug, while maybe not as deflectable should be just as ineffective against a lightsaber as a blaster shot is, due to the nature of a . That is, a blade made of plasma should simply vaporize the slug, should the slug manage to penetrate the containment field.     The weapon consisted of a blade of pure plasma emitted from the hilt and suspended in a force containment field. The field contained the immense heat of the plasma, protecting the wielder, and allowed the blade to keep its shape.   While a slug would seem to be more accurate than a blaster bolt (at least when used by a skilled marksman), I can't imagine that the slug would be faster than a bolt. Or if it is faster I wouldn't imagine it would be so fast that a Jedi would be unable to react with enough speed to block it, either with their saber or with the force. So what is the implied advantage of a slug thrower, versus a blaster, when used against a Jedi? Quotes insert problem while working with webservice quotes are not inserting may i know what i have to do. my code is     for(integer l=0;l&lt;reqquotes;l++){          qu = new Quote();          qu.Name = qName;          //opp.CloseDate =system.today();           qList.add(qu);             insert oppList;          insert qList;     } How to get rid of the smell from the fridge? What the #$@&%*! is that called? Is there a name for the use of symbols in place of curse words, for example #$@&amp;%*!? I've been testing PCA via SVD to decompose a simple time series data matrix, $X$.  I have two signals $x_1(t)$ and $x_2(t)$ in a data matrix where $M$ rows represents each timepoint sample and each column represents $x_1$ and $x_2$.  The mean signal, $\hat{x}$, is defined as the mean along the row axis (average of $x_1$ and $x_2$ along each timepoint).  I normalize each column of $X$ by subtracting its mean and dividing by the standard deviation.  When I use [U S V] = svd(Xz) in matlab, regardless of how the variables are distributed (whether they are correlated or uncorrelated), one of the columns of the right singular matrix, V, always points in the same direction (to a multiplicative constant) as the mean vector $(1/2, 1/2)$.  But when add an additional third time series, this is never the case (where the mean vector is $(1/3, 1/3, 1/3)$.  Because I normalize the standard deviation for each vector, it does make sense that the direction of most variance given by PCA would be the diagonal 45 degree line.  But if both variables $x_1$ and $x_2$ are independent gaussians, couldn't the PCA direction be any direction since the distribution is radially symmetric?  MATLAB Code:  s = RandStream('mcg16807', 'Seed', 0); RandStream.setDefaultStream(s);  G = zeros(1000,2); G(:,1) = 40*randn(1000,1)-100; G(:,2) = tan(G(:,1)) + randn(1000,1); X = G - repmat(mean(G),[size(G,1), 1]); Xz = X./repmat(std(G),[size(G,1), 1]); [U S V] = svd(Xz); G_mean = mean(Xz,2); corrcoef(G_mean, U(:,1)) corrcoef(G_mean, U(:,2)) The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural? Execute a command once per line of piped input? I want to run a java command once for every match of ls | grep pattern -. In this case, I think I could do find pattern -exec java MyProg '{}' \; but I'm curious about the general case - is there an easy way to say "run a command once for every line of standard input"? (In fish or bash.) How to resolve BackgroundContainer.dll error on startup? When I start my desktop it gives me an error about BackgroundContainer.dll.    How to get rid of it?  I found nothing when i started regedit.exe and searched (F3 or CTRL+F) for BackgroundContainer.dll Cleaning ‘staining’ off bottom of MacBook screen A couple months ago, I started to see some very small stain spots on the bottom of my MacBook Pro early 2015 screen. It's very annoying to see while I'm using it. When the screen is turned off, I see it as in the picture below. But when it's turned on, I see it as very small water-ish drops. I tried to clean it using water, screen cleaner, and even alcohol, but it didn't go away.  Also that area doesn't feel as smooth as the unaffected area. So, I think it's on the outside of my screen not inside.  Have you ever seen this before? Any suggestions on how to remove it without the need of replacing the whole screen? I tried Office 2013, but now wish to uninstall it. I tried to uninstall it but it always hangs at step  2 out of 4. It uses no CPU and no disk activity. So basically it's just idle.   Does anyone have a solution to this problem? This week I've read two news stories about two different camera lenses that can magnify things that are very far away.  The first is the , which is a point-and-shoot camera that costs $600, has an 83x optical superzoom lens ("the same field of view as a 2,000mm lens on a full-frame camera"), and lets you see the craters of the moon.  The second is the , which weighs 36 pounds, costs $180,000, and can take portraits of people a half mile away.  A non-photographer might ask, "Why would somebody spend their life savings on a gigantic telescope of a lens when you could take far away pictures with a handheld camera for 1/300th the price?"  Well, I am basically a non-photographer, and I would like to ask that question. What does the big one give you that the little one doesn't provide? I understand that small changes in lenses can make a big difference in size and price, but this seems to be of a different scale entirely.   What is going on with the optics of the lenses that makes the small one able to be so small and the big one have to be so...big?   I'm trying to get a basic intuition for what makes lenses different, and I realize I don't understand enough about how lenses work to even know how to compare these apples and watermelons. What are reserved filenames for various platforms? Why do some people use $+\infty$ instead of $\infty$? Why do some people use $+\infty$ instead of $\infty$? Because we usually denote minus infinity as $-\infty$, I think it is sufficient to denote plus infinity as $\infty$. Are there any reasons for this notation? I saw that many people majoring analysis use like $\int_0^{+\infty}$ or $\sum_{j=1}^{+\infty}$ or $\bigcup_{j=1}^{+\infty}$ which is not related to the unsigned infinity. How come I can't give more than 500 rep? Why can't I give more than 500 reputation as a bounty at a time? The system won't let me choose an option more than 500.  Also, how were some users able to do so? On  the OP has given a single bounty worth 550 rep, but at the moment I have 844 rep and I can't give any more than 500 rep on a bounty.  Can you please tell us if this is a bug or not? Visiting Greece with back-to-back Schengen visas (One from Germany, other from Italy)