Let $R$ be a commutative ring such that for each $a\in R $ there exists a positive integer $n&gt;1$ (depending on $a$) such that $a^n=a$. Prove that every prime ideal is a maximal ideal.   My try: Let $P$ be a prime ideal. Let $P\subset Q \subseteq R$. Then there  exists $q\in Q $ but $q\notin P$. Then $\exists n$ such that $q^n=q\implies q(q^{n-1}-1)=0\in P$ since $P$ is prime either $q\in P $ or $q^{n-1}-1\in  P$.  Thus $q^{n-1}-1\in  P$. How to complete the proof from here?
Let R be a commutative ring with an identity such that for all $r\in$ R, there exists some $n&gt;1$ such that $r^n = r$. Show that any prime ideal is  maximal. (Atiyah and MacDonald, Introduction to Commutative Algebra, Chapter 1, Exercise 7.)    Any hints?
I have a question on my assignment asking me to prove that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$. What I did so far is:  We know that $A\cdot A^t = I$ (since it is orthogonal), and that would mean that if $A$ is a matrix with elements $a,b,c,d$ then :  \begin{cases}a^2 + b^2 = 1\\ ac + bd = 0\\ c^2 + d^2 = 1\end{cases}  which implies that \begin{align}\det(A)&amp;= ad - bc\\ &amp;= -\frac{bd^2}{c} - cb\\ &amp;= -\frac{bd^2}{c} - \frac{c^2b}{c}\\ &amp;= -\frac{bd^2 + bc^2}{c}\\ &amp;= -\frac{b(d^2 + c^2)}{c}\\ &amp;= -\frac{b(1)}{c}\\ &amp;= -b/c\end{align}  I might be doing something wrong here, could someone please help me out understand this? An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm I'm writing a sports management simulation game. Like most sport management games, it makes heavy use of statistics. I would like to use real players, teams and league names.  i) Am I running the risk of getting into legal battles if I should use the real names? ii) Also what about team colours and emblems etc?    Update Much thanks for all the input! What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? If I'm not wrong, torque is perpendicular to both the radius and force i. e. It is along the axis of rotation. Questions that arise are- why do we consider the length between the axis/point of rotation while calculating torque? More importantly why is torque a cross product? How can I get a list of all the implementations of an interface programmatically in Java? Can I do it with reflection or something like that? Identical creatures entering the battlefield If I have two cards called  in hand and I play my first, Sin Collector (A). I use its triggered ability.   "When Sin Collector enters the battlefield, target opponent reveals his or her hand. You choose an instant or sorcery card from it and exile that card."  Now, I play Sin Collector (B), I use B's triggered ability.  But now my question. Does Sin Collector (A)'s ability trigger again as well? 'Sin Collector' entered the battlefield again, right? GSC: How to move a site from a domain to sub domain, sub folder or new domain with a sub folder? Is it possible to tell Google's Webmaster Tools that our site hosted on www. has moved to a sub folder, sub domain or a new domain with sub folder? Is Stack Exchange / Stack Overflow available for private or internal use? Is the Stack Exchange engine that powers Stack Overflow available to download or install internally for an enterprise or company?  I think Stack Exchange's engine is very great and could be very cool to use for internal enterprise patterns and practices, like the engine of Wikipedia. which analysis fit the best for the data? I am trying to work on the data of my master thesis. I have to create a model, but I don't know which analysis perform. My data look like this     number ID sp ab alt SWdiv     1 A1 4 6 630 0.25802965   2 A2 9 16 750 0.29235324   3 A3 8 14 850 0.411620203   4 A4 9 22 950 0.444079553   5 A5 2 12 1050 0.294855765   6 A6 13 40 1130 0.298451014   7 B1 2 2 750 0.275329152   8 B2 5 6 850 0.043352118   9 B3 14 16 950 0   10 B4 17 58 1050 0.230358688   11 B5 7 9 1150 0.298759725   12 B6 8 68 1250 0.012192474   13 C1 12 20 830 0.467542842   14 C2 13 19 930 0.454480722   15 C3 6 30 1030 0.464584038   16 C4 7 14 1130 0.448191833   17 C5 5 15 1250 0.052045275   18 C6 2 7 1315 0.051767368   19 D1 3 3 650 0.294855541   20 D2 11 17 750 0.291717599   21 D3 9 46 850 0.084284864   22 D4 8 57 950 0.088574421   23 D5 4 32 1050 0.160873283   24 D6 13 22 1150 0.297638962   25 D7 6 22 1250 0.290733158   26 E1 11 23 800 0.250878932   27 E2 8 8 900 0.418021272   28 E3 10 13 1000 0.132159303   29 E4 5 10 1100 0.058487554   30 E5 5 15 1200 0.250310822   31 E6 10 19 1350 0.01600807   32 F1 6 20 800 0.462497319   33 F2 15 54 900 0.376552904   34 F3 6 13 1000 0.298264791   35 F4 6 13 1100 0.394736812   36 G1 14 22 830 0.391455956   37 G2 11 16 930 0.282922551   38 G3 5 10 1030 0.282398234   39 G4 10 15 1130 0.153502726   40 G5 13 18 1230 0.051586706   41 H1 10 11 700 0.3009609   42 H2 7 14 800 0.579228102   43 H3 5 11 900 0.528108703   44 H4 20 48 1000 0.28689587   45 H5 13 18 1100 0.264542772   46 H6 10 20 1250 0.29928462   47 I1 12 13 535 0.35326876   48 I2 3 5 635 0.273947657   49 I3 18 30 720 0.467208221   50 I4 16 188 835 0.276669711   51 I5 7 69 935 0.000867067   52 L1 6 10 435 0.36910895   53 L2 7 20 606 0.236700524   54 L3 14 33 697 0.111132519   55 L4 17 42 797 0.296757045   56 L5 5 7 910 0.27539552   57 L6 10 15 1000 0.29748451   58 L7 6 11 1101 0.276755556     58 sample points each has two value: altitude and diversity index related to habitat heterogeneity. What I sample are different insects and I have two values: sp = species (total number of different species for each sample point) and ab = abundance (total number of insects for each sample point). My question are are the species richness or the abundance affected by the altitude or the habitat heterogeneity? which is the best analysis to perform? and why? thank you for your help. I got a visa with my mother when I was 17 years old. On it was stated "Not valid unless accompanied by my mother".    Can I travel to the UK alone now that I am 18? It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent. How long does the Truffle take to move in after building the mushroom biome? How do I find my follower if and when they leave me? The intuition I have for why quantum computing can perform better than classical computing is that the wavelike nature of wavefunctions allow you to interfere multiple states of information with a single operation, which theoretically could allow for exponential speedup.   But if it really is just constructive interference of complicated states, why not just perform this interference with classical waves?  And on that matter, if the figure-of-merit is simply how few steps something can be calculated in, why not start with a complicated dynamical system that has the desired computation embedded in it. (ie, why not just create "analog simulators" for specific problems?) How can I send an email using a template? I have created a email template in ExactTarget API and I want to use it executing XML from SoapUI.  Is there any tutorial or examples of how can this be done?  I'm new at this and I'm not sure how ExactTarget really works. "Contact us"... and then what? So I had a dispute with the moderators at a specific SE site. As a result, I got suspended there. As per SE guidelines, I pressed the "Contact us" button and wrote to the higher powers to appeal.  That was 5-6 weeks ago. There was zero feedback, neither positive, nor negative, not even "it's in progress, please wait".  So what is this "contact us" thing supposed to do? Is it a placebo to make me feel better while being routed to dev/null?  What is the normal time it takes to process a "contact us" message?  Notes:   As far as I remember, I did not contact the team before. I'm a longtime user of multiple SE sites and never had any problems except for that one site. I'm pretty sure my form was submitted. I didn't make screenshots or anything, because I did not think I would need them in a post like this. I didn't expect a super-fast reply either, so I wasn't wondering after a week or two. Can I have two Schengen visas in one passport? What did the public at large know about the assassination attempt on Palpatine and how he survived it? Safari can load internet but none of the other apps cannot! This is by far the weirdest problem I've come across,  My mac can perfectly access internet through safari web browser, but when I tried to do   ping www.google.com   through the terminal, it's requests is being timed out.  None of the other apps are able to reach internet either. Dropbox, Skype, Spotify all showing offline, except the browser itself  I'm guessing something bad with the DNS, but spend 2 hours trying to fix this with no luck