Let $A$ is a commutative ring with $1$ and $a$, $b$ are the elements of $A$ such that $a+b=1$ and there is some $n&gt;0$ , $(ab)^n=0$. Prove: $(1)=(a^n)+ (b^n)$.    I would say it's kind of easy one, but somehow I got stuck on it too long. Please help me !
Possible Duplicate:          On  it's a stated fact that if $I$ and $J$ are coprime ideals of a ring $A$, then $I^k$ and $J^\ell$ are also coprime for any integers $k,\ell$.   There's no proof given. Why is this fact true?
Maximal ideals in the ring of real functions on $[0,1]$ Let $S$ be the ring of all continuous functions from $[0,1]$ to $\mathbb R$. How to prove that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$?   Thanks in advance. I am wondering how to differentiate all these different joins ... How do I fix this "E: Type '*' is not known on line * in source list ..." update error? Every once in a while I see users having issues to update due to errors like this:  E:Type 'ain' is not known on line 1 in source list /etc/apt/sources.list.d/some-ppa.list'   with varying types/line numbers/source list files (often after removing a PPA).  How can such an error be fixed? I am working on a proof about Dedekind cuts on the positive rational numbers. I have been stuck for a while on the following point and would appreciate any help.  Given $x\in \mathbb{Q}^+$ such that $x^2&lt;2$, how to find $y\in \mathbb{Q}^+$ such $y&gt;x$ and $y^2&lt;2$? How to generate all permutations of a list? Changing CSV layer attribute value from text to number in QGIS? I read in a CSV file for linking to a .shp file.   All the variables that are just numbers came in as text.   How do I change the text variables to numeric in QGIS? What is the origin of "a cut above/below the rest"? Where did the phrases "a cut above the rest" and "a cut below the rest" come from? I'm supposing they came from the same source. I feel it something very obvious, but I don't know what. Why can't I change directories using "cd" in a script? Temporarily deleted answer missing from asker's inbox I  and . This answer does not appear in my inbox. I never saw a notification, and as of now the latest notification from CS is  {"SiteUrl":"cs.stackexchange.com","FaviconUrl":"https://cs.stackexchange.com/favicon.ico","CreationDate":"17 hours ago","Url":"https://cs.stackexchange.com/questions/641/language-of-the-graph-of-an-affine-function/667#667","Title":"Language of the graph of an affine function","Summary":"Hint:\n\n$53 \\cdot 100000010000000000 + 4 = 5300000530000000004$.\n\nSketch:\n\n\n  ! Assume $a$ has $n$...","Type":"answers","Count":3,"IsNew":false}   This answer was posted, deleted and undeleted while I wasn't online. I suppose the deletion removed the entry from my inbox. Then the undeletion should have recreated an entry but didn't. So I am running on Windows 7 (64-bit) Home Premium with 12 GB RAM (Intel Core i7 920).  I have restricted the Page File to 400MB since it was eating up a lot of space om my SSD (that is 80GB).   After that I sometimes get the "low memory"-warning, like this, except for the behold-comment:   When I get those warning I have usually a lot of RAM available; when I check Resource Manager I have over 2GB "Free" and over 2GB "Available" - usually more than that.  The diskspace on all my drives have over 10GB free.  So the question is - why does Windows complain? It actually restarted by computer (hard boot) when it happened, and as I said - lots or RAM available. Ubuntu boots on terminal only, how do I recover the GUI desktop? I want to make a plane for my supply on my PCB and a plane for my MCU to isolate the MCU from the power supply noise.  I found this article in Altium designer :  But it isn't clear how the split plane tool work. Can someone explain it in a different manner?  [EDIT]  This is a picture how I see the planes: "A friend of my father" vs. "a friend of my father's" — which one is right? Why? I have seen them both in my reference book, which is written by a native speaker. I wonder why there are two different phrases to describe the same meaning and am looking forward to some reasonable details to explain why this is so. From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question.   Would it be right to use some idea of a splitting field here? Or is it something to do with $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ not containing a primitive 3rd root of unity?   This is a past exam paper question worth 8 marks, so it doesn't seem like it's a few lined answer, but I am a little stuck on where to go. I was using this command to use a custom crafting table:  /execute @a ~ ~ ~ detect ~ ~-1 ~ dropper 1 {Items:[{id:minecraft:blaze_rod,Slot:1b,Count:1b,Damage:0s},{id:minecraft:blaze_rod,Slot:4b,Count:1b},{id:minecraft:stick,Slot:7b,Count:1b}]} /scoreboard players set @p LSC 1   But I get this error message when I stand on a dropper with these items:     Failed to execute '{Items:[{id:minecraft:blaze_rod,Slot:1b,Count:1b,Damage:0s},{id:minecraft:blaze_rod,Slot:4b,Count:1b},{id:minecraft:stick,Slot:7b,Count:1b}]} /scoreboard players set @p LSC 1' as TrevMCC38   By the way, Im in 16w06a Wikipedia claims this but only cites an offline proof:  $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$  for $n$ square roots and one minus sign. The formula is not the "usual" one, like Taylor series or something like that, so I can't easily prove it. I wrote a little script to calculate it and it's clearly visible, but that's not a proof. Identification of mid-80s time travel tv show I've sectioned of this area in the sea near my base, and I need to get rid of all the water from it.  From what I've tried the only way I can see to clear it is fully cover one layer, delete the blocks, then go down to the next layer. Is that the only way I can clear the area or is there a better approach?  Ideally I'd like a survival method (if there is a better one), but any commands what could be used I'd like to hear just so I can learn more commands :) Let $A$ be a real square matrix en let $I$ be the identity matrix with the same size. Let $n&gt;0$ be the smallest natural number for which there exist  $a_0,a_1,...a_{n-1} \in {R}$ such that $A^n+a_{n-1}A^{n-1}+....+a_1A+a_0I=O$.  Show that A is invertible if and only if $a_0\neq0 $.  So i need to proof    $A$ invertible $\rightarrow$$a_0 \neq 0$ $a_0 \neq0 \rightarrow$ $A$ invertible    I don't really know where to start with 1, and with 2 I tried to prove it with a contradiction but I don't think it's right either. How to install gnome shell extensions offline? I know how to go to the  website and download gnome-shell extensions, but now I need to install some extensions available there on a computer without any internet access at all. It is in a internal corporate network and there's no way I can get outside internet access on it, so I must find another way. I can copy files in a usb disk.   At my home computer, I have found my extensions at ~/.local/share/gnome-shell/extensions/ but just copying this folder to the target corporate computer didn't do the trick.   Running gnome-tweak-tool gives me a "Install Shell Extension" button but I don't know how to download an extension in a format acceptable to install using this button. I have tried to point to the folder above but it didn't work either.  What do I need to do?