Let $R$ be a commutative ring with identity.For each $a\in R$ there exist $n(&gt;1)\in \mathbb N$ such that $a^n=a$.   Prove that every prime ideal of $R$ is maximal.  My try  Let $I$ be a prime ideal of $R$. Then $R/I$ is an integral domain. If I can show using the hypothesis that $R/I$ is a field then we are done.  Any hints to show this?
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?
How does RAID 1 determine which drive should act as the source for a rebuild? Here's my scenario -- both drives were working perfectly fine. I removed one of the drives for a while (building a new PC, didn't think I was going to use RAID anymore but changed my mind). I actually switch the SATA type to AHCI and was operating for about a week on the new system before deciding to re-enable RAID and reinstall the drive I had removed.   It's currently rebuilding, but I'm curious how it knows which one is the source and which one is out-of-date? I am solving a clustering problem on a trivial dataset with the k-means algorithm. I am running a couple of tests and, from time to time, one centroid doesn't attract any points, i.e. I've got an empty cluster (see the purple "x" in the picture).  What should I do? Shall I delete it or just stop updating its value? Why?    I am aware that built-in functions (e.g., kmeans() in R) have automatic ways of dealing with this situation, but I am trying to write the standard algorithm from scratch. As soon as I fix it I'll be able to compare my results to built-in functions. At this moment I'm looking for some theoretical reasons why I should prefer one solution or another.  In the picture each colour represents a cluster according to the current iteration and each "X" is its centroid (old ones have been kept and marked with the number of the iteration in which they were computed).      [Some related links about empty cluster case, added later by @ttnphns: ; ; ; ; ; ; ; ; ] What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? Can I install Python 3.x and 2.x on the same Windows computer? Reviewing suggested edit of own post after completing 20 reviews I've completed 20 suggested edits reviews today. Then, I came across a  and I approved it.   Then, I had a doubt that whether the review count has incremented or not, oh! it's not incrementing. System works fine.  But, take a look at review page,     The edit was already reviewed by one reviewer and then I reviewed it. As I'm the Owner of the post it's enough to approve the edit. But, the review history doesn't show the detail that I had approved it. It's misleading that only one vote of a reviewer who is neither the community user nor the OP nor the moderator, is enough.  So, what should we do in this type of situation?  Is it correct to make OP to review the suggested edit after completing daily limit?  Should we need to display/hide the OP has approved the edit in the review history? For bash script, I can use "$@" to access arguments. What's the equivalent when I use an alias? Blending colors in photoshop and illustrator cd to previous directory without echo of directory name This  gave me cd - as shorthand to change to my previous directory in Bash. It does have a major problem: it prints the directory it changes to, wasting my screen space, as that directory is then repeated in the prompt on the next line.  I know I can do cd $OLDPWD, also mentioned in that answer, and cd - &gt;/dev/null but that is much more typing.   In the bash man page it says:     ... or if - is the first argument, and the directory change is successful, the absolute path name of the  new  working  directory  is written to the standard output.   How can I make - not to be the first argument? By using any of the cd optional arguments? Any other way to suppress the echo or minimise typing without the need for defining functions/aliases on every machine I work with? Helpful flags not getting updated after getting 'close question' privilege The novel centers around a man living a comfortable existence in the pretty distant future.  To support himself, he is occasionally asked to enter a box which will copy his entire body, and send it out faster than the speed of light to some developing world, to work.  His copy on this world wakes up and realizes what a big mistake he's made.  The copy does grueling work, feels lonely, and eventually just dies of radiation sickness.  Back on earth, the man never realizes how bad it is out there, and keeps sending out more copies.  I think the story is probably 80s or earlier - not sure if it's a short story, or the beginning of a novel which I lost &amp; never finished. I'm setting up an Nginx web server and and want to perform load balancing using the round robin method.  This is what I have found thus far:  upstream backend  {    server backend1.example.com;    server backend2.example.com;    server backend3.example.com; }   According to the page they says i need to add the above lines to my nginx.conf file and upload it to server to load balance.  I do not understand what it means by server backend3.example.com;.  How do I setup a backend instance? Number of possible combinations of x numbers that sum to y I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.  An asymptotic approixmation is also useful. This question seems to be very close to number partitioning, with the difference that a number can be 0. See:    All possible partitions for sum 10 and 3 positions that can be zero, are 63 possiblities: (numbers shown as 3 digits)  019 028 037 046 055 064 073 082 091 109 118 127 136 145 154 163 172 181 190 208 217 226 235 244 253 262 271 280 307 316 325 334 343 352 361 370 406 415 424 433 442 451 460 505 514 523 532 541 550 604 613 622 631 640 703 712 721 730 802 811 820 901 910 How accurate is the location detection on the iPhone's Find My iPhone app Custom kernel: fails to load firmware when module built-in I'd like to have all my modules built-in, but this fails with iwlagn:  iwlagn 0000:03:00.0: request for firmware file 'iwlwifi-6000-4.ucode' failed. iwlagn 0000:03:00.0: no suitable firmware found!   The microcode file exists in /lib/firmware and the whole thing works just fine if I compile iwlagn as module. I have no idea where it's looking for the file or what's wrong - any ideas? I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$. I suspect, that there are no more solutions. The problem is that I don't know how to prove that.  Since we have a rational equation, I have no idea how to make any substitutions in order to get expression for $\tan x$ (as it is not a rational expression).  P.S. I can also show that it is true that (a) $f(0) = 0$ (take $x=y=0$), and (b) $f(-x) = -f(x)$ (take $y=-x$).  Update: there is a solution of this problem also here: Fundamental group of a torus with points removed Is it really possible to break the speed of light by flicking your wrist with a laser pointer? How to instruct Windows 8 NOT to perform a fast shutdown? I have enabled fast boot in my copy of Windows 8 but I would like to be able to specifically instruct Windows not to perform a "fast shutdown" whenever I want (without disabling fast boot altogether).  I need to be able to do this because I dual-boot Ubuntu and therefore can't mount an NTFS partition as read/write if Windows has performed a "fast shutdown". (Well okay, I can force the partition to be mounted read/write, but then I could run into data loss/corruption problems.) I am developing the backend for a website. I always had the trouble of making an attractive design for my websites and have never hired any graphic designer yet. I have no logo nor web template.  What are the steps to hire a graphic designer? What information should I read about before hiring one in order to make most out of our time? Where should I find him? What questions should I ask him? How much do they charge and what qualifications should I look out for? Should I disclose my business ideas to the designer or could that be risky? A book suggests that we should not interchange "so" and "so that". "So" means "therefore", and "so that" means "in order that". However, it seems to me that in many cases they don't actually have a difference. For example:     Alice got up at 5:00 A.M., so that she could drive her son to school.   If I substitute "so that" with "so", this sentence presents the exact same causal relation to me:     Alice got up at 5:00 A.M., so she could drive her son to school.   I must have missed something here, since English is my second language. Could somebody explain the difference?