There's an exercise in Herstein marked as very hard     Let $R$ be a commutative ring. If $q(x) \in R[x]$ be a zero divisor in $R[x]$, then if $\displaystyle q(x) = \sum_{0 \leq i \leq k} a_i x^i$, prove there's $b \in R$ such that for all $0 \leq i \leq k$, we have $ba_i = 0_R$    Now how I am supposed to solve this ?  I know that I'm supposed to prove how much I have worked by myself before asking for help, but I tried for more than two hours but still can't even solve the case when $\text{deg}[q] = 1$
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 $ba_0=ba_1=\cdots=ba_n=0$?
Specifying spaces between words How do servlets work? Instantiation, sessions, shared variables and multithreading Illustrator Vector Graphics Appearing Pixellated This has been puzzling me for a few days now, and I asked a graphic designer I knew before trying here and haven't been able to solve it yet. It's quite weird so please bear with me for a bit.  I am using Adobe Illustrator to create a logo, and I have something that can potentially work. I have used only the pen tool with strokes (no fill), but the issue is, when I view the logo at 100%, it looks pixellated! I know vectors should not even be able to appear pixellated, but they are. Here is the screenshot from AI, without exporting to anything:     That was just done using the snipping tool within Windows, and it's at 100% zoom.  I have anti-aliasing turned on from Edit > Preferences > General, and the document resolution is at 300 ppi (I get the same result with 72 ppi).   Could anyone help me with this?   If relevant: Using CS6, and have a 24 inch Dell U2412M at 1920x1200 resolution. I do have a 21.6" Samsung at 1680x1050 and I get the same result there as well. Let me know if you need any other settings, or if you would like the file, I'd be happy to provide it to be able to solve this problem. How do you find the Lie algebra of a Lie group (in practice)? Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition. Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field. I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one please check this proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field. Thank you.  Proof: Notice that $$\langle 1 - i \rangle\\  \Rightarrow 1 = i\\  \Rightarrow 2 = 0.$$ Thus all elements of the form $a+ bi + \langle 1 - i \rangle$ can be rewritten as $a+ b + \langle 1 - i \rangle$. But since $2=0$ this implies that the elements that are left can be written as $1 + \langle 1 - i \rangle$ or $0 + \langle 1 - i \rangle$. Thus  $$ \mathbb Z[i]/ \langle 1 - i \rangle = \{ 0+ \langle 1 - i \rangle , 1 + \langle 1 - i \rangle\}. $$  This is obviously a commutative ring with unity and no zero-divisors, thus it is a finite integral domain, and hence is a field. $\square$ C# vertical label in a Windows Forms Do I need a Schengen visa if I have UK permanent residency? Halo particles don't render in Cycles I'm writing a java app using eclipse which references a few external jars and requires some config files to be user accessable.   What is the best way to package it up for deployment? My understanding is that you cant put Jars inside another jar file, is this correct? Can I keep my config files out of the jars and still reference them in the code? Or should the path to the config file be a command line argument? Are there any third party plugins for eclipse to help make this easier? I'm using an ant build file at the moment but I'm not sure I know what I'm doing. Is there an equivelent of the deployment projects in Visual studio, that will figure out everything you need and just make an installer? I've used install4j before, and it was powerful if no where near as automated as  .Net deployment projects.   Cheers. Can we some how use color labels on icons like OS X?  Something like this: More examples of Simpson's Paradox, barring the ones on Wikipedia, Titanic, and delayed flights. Can I open Microsoft Word and Excel files with Ubuntu? Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.   This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.   (Please note that $e$ in the question is the group's identity.)  Here's my attempt though...  First I understand Abelian means that if $g_1$ and $g_2$ are elements of a group $G$ then they are Abelian if $g_1g_2=g_2g_1$...  So, I begin by trying to play around with the elements of the group based on their definition...  $$(g_2g_1)^r=e$$  $$(g_2g_1g_2g_2^{-1})^r=e$$ $$(g_2g_1g_2g_2^{-1}g_2g_1g_2g_2^{-1}...g_2g_1g_2g_2^{-1})=e$$  I assume that the $g_2^{-1}$'s and the $g_2$'s cancel out so that we end up with something like,  $$g_2(g_1g_2)^rg_2^{-1}=e$$ $$g_2^{-1}g_2(g_1g_2)^r=g_2^{-1}g_2$$  Then ultimately...  $$g_1g_2=e$$  I figure this is the answer. But I'm not totally sure. I always feel like I do too much in the pursuit of an answer when there's a simpler way.  Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra. Number Theory: Prove that $m | \phi(a^m - 1)$ I am relatively new to game development, and am working on a simple 2D platformer. I have created a tilemap and painted some tiles for the character to walk on.. but he is getting stuck. Yes, I know this is a common question and I did find some answers (including this one: ), but none seem relevant for me. The thing is my tiles are not flat.. they have pointy edges due to grass.. see the image below:  As you can see, the edge is not flat. I have overcome the problem of getting stuck by changing the &quot;Edge Radius&quot; to what you see in the above screenshot and that works, but the character now looks like he is floating slightly above the ground. Is there any way I can just tell Unity to ignore the jagged grass and make the collider flat and move the edge of it down a bit? That way it would fix the issue and also have the added bonus of looking like some of the grass is behind him and some in front.. Looking for a scifi novel with advanced pheromone technology What is "#!' in a script file? $cat shebang.sed  #! /bin/sed -f s/red/xxx/ s/BLUE/1234/  $./shebang.sed rgb lower (#1): "xxx green blue" UPPER (#2): "RED GREEN 1234"  $sed -f shebang.sed rgb lower (#1): "xxx green blue" UPPER (#2): "RED GREEN 1234"   I may guess what #! is doing. However, I don't say I know it. what is "#!"? Possible Duplicate:    When we tell a joke or narrate a story in simple present can we use 'would' in some cases? For example:  He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he will take revenge He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he would take revenge The Riemann zeta function $ζ(s)$ is defined for all complex numbers $s ≠ 1$ with a simple pole at $s = 1$. It has zeros at the negative even integers, i.e., at $s = −2, −4, −6, ...$.   My question: How one can obtain these roots. Can I watch the progress of a `sync` operation?