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Let $A \subset B$ be a finite ring extension. Fix a ring homomorphism $\nu : A \to \Omega$, where $\Omega$ is an algebraically closed field.   I want to show there exists a non-zero homomorphism $v : B \to \Omega$ of $A$-modules.  I want show that the space $V$ of all the homomorphisms $v$ in (i) is a non-zero finite dimensional vector space over $\Omega$. Show that the algebra $B$ acts on this space. Show that eigenvectors of this action correspond to homomorphisms of algebras $\lambda : B \to \Omega$ that extend the homomorphism $\nu$. Deduce that the set of such homomorphisms $\lambda : B \to \Omega$ is finite and non-empty.    We can assume that $B = Ab_1 + ... + Ab_n$ where $b_1,...,b_n \in B$. The chose $v(b_i)=\omega_i \in \Omega-\{0\}$ and then define $$ v( a_1 b_1 + ... + a_n b_n ) = \nu(a_1) v(b_1) + ... + \nu(a_n) v_n = \nu(a_1) \omega_1 + ... + \nu(a_n) \omega_n \ .$$ This is an $A$-module homomorphism. Am I correct? For the second part, that $V$ is not-zero follows from the first part. That it is finite dimensional follows from the fact that $v_i(b_j)=\delta_{i,j}$ (Kronecker's delta) with $i=1,...,n$ spans $V$ and hence $\dim V \le n &lt; \infty$. I define the action of $B$ over $V$ to be $$b \bullet v = \left( x \mapsto v(b)v(x) \right) \mbox{  for all } b,x \in B \ .$$ This seems to be the natural action to define. Am I correct? However, I haven't managed to solved the rest (the eigenvector stuff and the relation to $\nu$). Why does the question's closed status disappear when clicking to load a new edit? I've come across a small issue which occurs when a question has been closed but the question itself or one of its answers are being edited.  Steps to reproduce:   Visit a closed question. (even closed years ago)   In a different tab or browser, edit the question or one of its answers. "an edit has been made to this post; click to load" message will appear above the edited post in the first tab/browser, click it. The .question-status div becomes hidden, which includes the closed notice below a question and, for a duplicate-closed question, the list of duplicates above the question body. This gives the appearance that the question is no longer closed (until you refresh).   Some screenshots:                I know it isn't common/important but it did confuse me when looking at a particularly bad question - I thought it had been reopened.   Is this an intentional feature? If so, what was the thinking behind it?    Another example  Before    After What is the easiest way to encode a PHP string for output to a JavaScript variable?  I have a PHP string which includes quotes and newlines.  I need the contents of this string to be put into a JavaScript variable.  Normally, I would just construct my JavaScript in a PHP file, à la:  &lt;script&gt;   var myvar = "&lt;?php echo $myVarValue;?&gt;"; &lt;/script&gt;   However, this doesn't work when $myVarValue contains quotes or newlines. Example of non-flat modules Let $R = \mathbb{C}[t]$ be a ring of polynomials in variable $t$ with coefficients in the field of complex numbers $\mathbb{C}$ and let $$N = R[x]/(tx-t).$$  I claim that $N$ is not a flat $R$- module. If we consider the exact sequence $$0 \rightarrow(t) \rightarrow R$$ such that the ideal $(t)$ is viewed as an $R$-module.   We know that the above sequence exact iff the map $(t) \rightarrow R$ is injective. Let us assume that N is flat then this means that $N \otimes  (t) \rightarrow N \otimes R$ has to be injective. How do I go from here?   Thanks. How to use Terminal to delete all .svn folders recursively? What command can I type into the Terminal so that I can delete all .svn folders within a folder (and from all subdirectories) but not delete anything else? skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. Does the condition $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ imply that $a_n$ convergent?  When I say convergent I mean to $L\in \mathbb{R}$ or to $\infty$ or to $-\infty$  I have a gut feeling this statement is false, just can't find a series which contradicts it.  Thanks a lot Where are the default settings saved for the Command Prompt defaults?  Are they stored in the registry somewhere, or in a file?  I would love to make a .reg file to set my default preferences when logging into a machine for the first time.  I'm talking about the default application settings: QuickEdit mode, buffer size, text color, etc. I have a scenario where I have multiple threads adding to a queue and multiple threads reading from the same queue. If the queue reaches a specific size all threads that are filling the queue will be blocked on add until an item is removed from the queue.  The solution below is what I am using right now and my question is: How can this be improved?  Is there an object that already enables this behavior in the BCL that I should be using?  internal class BlockingCollection&lt;T&gt; : CollectionBase, IEnumerable {     //todo: might be worth changing this into a proper QUEUE      private AutoResetEvent _FullEvent = new AutoResetEvent(false);      internal T this[int i]     {         get { return (T) List[i]; }     }      private int _MaxSize;     internal int MaxSize     {         get { return _MaxSize; }         set         {             _MaxSize = value;             checkSize();         }     }      internal BlockingCollection(int maxSize)     {         MaxSize = maxSize;     }      internal void Add(T item)     {         Trace.WriteLine(string.Format("BlockingCollection add waiting: {0}", Thread.CurrentThread.ManagedThreadId));          _FullEvent.WaitOne();          List.Add(item);          Trace.WriteLine(string.Format("BlockingCollection item added: {0}", Thread.CurrentThread.ManagedThreadId));          checkSize();     }      internal void Remove(T item)     {         lock (List)         {             List.Remove(item);         }          Trace.WriteLine(string.Format("BlockingCollection item removed: {0}", Thread.CurrentThread.ManagedThreadId));     }      protected override void OnRemoveComplete(int index, object value)     {         checkSize();         base.OnRemoveComplete(index, value);     }      internal new IEnumerator GetEnumerator()     {         return List.GetEnumerator();     }      private void checkSize()     {         if (Count &lt; MaxSize)         {             Trace.WriteLine(string.Format("BlockingCollection FullEvent set: {0}", Thread.CurrentThread.ManagedThreadId));             _FullEvent.Set();         }         else         {             Trace.WriteLine(string.Format("BlockingCollection FullEvent reset: {0}", Thread.CurrentThread.ManagedThreadId));             _FullEvent.Reset();         }     } } What are the C++ coding and file organization guidelines you suggest for people who have to deal with lots of interdependent classes spread over several source and header files?  I have this situation in my project and solving class definition related errors crossing over several header files has become quite a headache. What's a big-vocabulary word for someone with a big vocabulary?  I'd like to say I'm looking for a "_____". I always wondered if children who graduate from Hogwarts have an education that is actually worth while in the real world.  Do the children learn math and English?  I can see History of Magic as practical. I know of Arithmancy, but that's the study of the magical properties of numbers, not math.   Since Hogsmead is one of the very last all-magical towns, it's a fair assumption that a majority of the students at Hogwarts will be living in non-magical towns or cities. If practical classes (like math or English) aren't taught, then how do these students function in the non-magical cities? Can SQL Server's Page Life Expectancy be too high? I've read several articles about SQL Server's Page Life Expectancy and what it means and what kind of information you can glean from it. In most of these articles that I've read, a common, healthy PLE value is somewhere around 1000-2000 seconds. I've read that getting down to around 300 seconds can mean you're likely low on RAM.  I recently just upgraded our hardware to have 64GB of RAM, up from 14GB. On the 14GB my PLE was around 300 seconds, and I was having a lot of 5-6 memory grants pending per second. So, that was bad and I increased the RAM. Now, my PLE is much, much higher around 5000 seconds, and no more pending memory grants. I've seen it 7000 seconds, if I recall. This is much higher than anything I've read.  Can a high PLE be a bad thing? Or is it the higher the better?  EDIT: I'm sorry, my PLE wasn't 7000 seconds, it was 70,000 seconds! Although, at the moment it is down to around 7000.     The database size is around 160GB. A few tables have 5+ million rows. max_server_memory is set to 2147483647. Thor: Ragnarok ends with the remaining people of Asgard aboard a refugee ship that meets Thanos' ship. Avengers: Infinity War opens with      Thanos and his minions having commandeered the ship and several bodies can be seen on the floor.   This begs the question, what happened to these characters:   Korg Miek Asguardian extra #3 Valkyrie I've connected a USB to Serial bridge dongle, and running dmesg | grep tty outputted the following:  [    0.000000] console [tty0] enabled [603199.380677] usb 2-2: cp210x converter now attached to ttyUSB0   So now I'm trying to write to it as per  by running the following:  cat hello.txt &gt; /dev/ttyUSB0   and   sudo cat hello.txt &gt; /dev/ttyUSB0   But both result in the following error:  bash: /dev/ttyUSB0: Permission denied   What am I doing wrong?  Thanks Poll on best VNC / remote desktop software for assisting others on Windows/Mac machines from Ubuntu?    I've heard good things about TeamViewer and Fog Creek Copilot, but I'm wondering if the included GNOME Vinaigre VNC client is good enough for this.  To specify, I'm looking for best option based on:   SIMPLEST ease-of-use for client to download/use on their end. See #1. Works cross-platform I am able to control client's mouse and/or keyboard from remote machine. Is it possible to apply modifiers to multiple objects at once? Sending Text Cross Domain By Bookmarklet How do I install Adobe Flash player? I have had significant problems with watching flash video in 64-bit Ubuntu.  Does anyone know of a good way to get flash running on the platform? Ubuntu App Development. Unable to install Ubuntu SDK and buggy Ubuntu emulator