$R$ be a ring and $a\in R$ be a non nilpotent element and define $S=\{a,a^2,a^3,\cdots\}$ and $F=\{A\subseteq R| A-ideal \;, S\cap A=\emptyset\}$ If $P$ is maximal ideal of $F$, is $P$ prime ideal? I am self studying abstact algebra but I didnot understand first of all is the set $F$ a ring? In what operations? Furthermore, how one can show this ?
In the midst of attempting to prove that for a commutative ring $A$ with identity, and an ideal $I$ of $A$, $I = \operatorname{rad}(I)$, where $\operatorname{rad}(I) = \{x: x^m \in I, m &gt;0\}$, implies that $I$ is an intersection of prime ideals, I've run into a little bit of confusion.   An answer to another question on this site, , indicates a strategy that I am attempting to use. I know that if $x$ is an element of $A\setminus I$, then so too is $x^m$ for any positive integer $m$. I use this fact to consider a subset $X = \{x^m : m &gt; 0\}$, which is closed under multiplication, and use Zorn's lemma to find an ideal of $A$ containing $I$ maximal with respect to the constraint that said ideal, $J$, is disjoint from X.  I want to prove $J$ is prime by contradiction. I suppose that $J$ were not prime; this also means that $J$ is not maximal in $A$. Then there would exist two elements, $f$ and $g$, of $A$ so that $fg$ is in $J$, but neither $f$ nor $g$ is in $J$. Following the prompting of the aforementioned answer, I know that $J + \langle a\rangle$ and $J + \langle b\rangle$ would be ideals strictly containing $J$, and since J is the maximal ideal disjoint from $X$, both $J +  \langle a\rangle$ and $J + \langle b\rangle$ contain at least some elements of $X$ (I am tempted to say all of them, but I'm not entirely sure that's true, in case for example $\langle a\rangle$ started containing powers of $x$ at $x^2$, in which case it isn't necessarily true that $\langle a\rangle$ contains $x$ itself), and so both $\langle a\rangle$ and $\langle b\rangle$ must themselves contain elements of $X$.  And this is where I am stuck. I feel like there is something I should know about multiplicative subsets of rings that would help me arrive at the desired contradiction, but I can't remember what it might be.
Improving Review Queues - Design overview II: Changing review bans and other new features I tried to create a script by echo'ing the contents into a file, instead of opening it with a editor  echo -e "#!/bin/bash \n /usr/bin/command args"  &gt; .scripts/command   The output:     bash: !/bin/bash: event not found   I've isolated this strange behavior to the .  $ echo ! !    $ echo "!" bash: !: event not found  $ echo \#! #!  $ echo \#!/bin/bash bash: !/bin/bash: event not found    Why is bang causing this?   What are these "events" that bash refers to?     How do I get past this problem and print "#!/bin/bash" to the screen or my file? How do I post code in chat? I would like to be able to post code in chat. I know that I can post inline snippets of code by using backticks, but I don't know how to post blocks of code, with the entire chat message formatted as code, and indentation respected. I am aware of . After reading the comments, I found out that I could format in "fixed width", but I can't see a "fixed width" button anywhere. What should I do? I just installed my first Ubuntu Server (12.04) instance and immediately ran into the following problems:   Running apt-get install or update fails (see below) Running ifconfig produces peculiar IP addresses starting with 10.x.x.x and my router does not see the server at all in its "Attached Devices" page (its a consumer grade Netgear)   When I run apt-get I get 50+ of the following errors:  Failed to fetch &lt;some-http-url&gt;. Temporary failure resolving &lt;blah&gt;.ubuntu.com   I read up on this error for 12.04, and found some similar posts on this forum and on server fault that recommended going into /etc/resolv.conf and adding an entry:  nameserver 8.8.8.8   To my surprise, I found that /etc/resolv.conf was a symlink pointing to somewhere else  and I had to delete it first, then create a new one with the entry indicated above.  I restarted the server and tried running apt-get again, same results.  How do I start diagnosing the problem (I'm a relatively new Linux/Ubuntu user)?  Additional details that may help:   This is actually a virtual machine running as a VirtualBox guest OS The physical host that is running this VM is my laptop which has a wireless connection; I'm wondering if (somehow) the laptop is getting the network wirelessly, but perhaps either the VBox VM and/or the Ubuntu server OS isn't configured for wireless and so nothing is "getting through"?   Thanks in advance for any pointers!  Edit:  Here is a snapshot of my VM's Network settings (only Adapter 1 has any information populating its fields, so I assume my VM has only 1 adapter):    Here is the output when I run ifconfig from the terminal:    And here is the output when I run route from the terminal:    Running ping google.com: How can one run multiple versions of PHP 5.x on a development LAMP server? CSS background opacity with rgba not working in IE 8 I'm using  and trying to switch from Web Applications to Stack Overflow.   I enter sta into "Switch sites" field, but search showing unrelated results first (3D Printing, 3D Printing Meta, Software Engineering ...): I have a financial transaction that looks something like this (pseudo):  begin     select record for update     update record(same record as selected)     insert another record into another table commit   If I call SELECT ... FOR UPDATE within the transaction, will the lock be released for the specific rows only when the transaction is committed ? I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$. I am currently in a clan with three members. When we tried to initiate a clan war it said we had insufficient clan requirements. Does anyone know what the full requirements are? Is there a minimum clan member requirement? "ImportError: No module named _struct" configuring python-minimal When I run sudo apt-get upgrade, the dpkg error appears and interupts the process.  Error details:  Setting up python-minimal (2.7.3-0ubuntu7.1) ... Traceback (most recent call last):   File "/usr/local/lib/python2.7/runpy.py", line 162, in _run_module_as_main     "__main__", fname, loader, pkg_name)   File "/usr/local/lib/python2.7/runpy.py", line 72, in _run_code     exec code in run_globals   File "/usr/local/lib/python2.7/compileall.py", line 16, in &lt;module&gt;     import struct   File "/usr/local/lib/python2.7/struct.py", line 1, in &lt;module&gt;     from _struct import * ImportError: No module named _struct dpkg: error processing python-minimal (--configure):  subprocess installed post-installation script returned error exit status 255 dpkg: dependency problems prevent configuration of python:  python depends on python-minimal (= 2.7.3-0ubuntu7.1); however:   Package python-minimal is not configured yet.   And this problem leads to a series of dependency problems when processing the following packages:    python  gwibber-service  libgwibber3  libgwibber-gtk3  gwibber  gwibber-service-facebook  gwibber-service-identica  gwibber-service-twitter  python-all  python-dev  python-all-dev  python-apt  python-problem-report  python-apport  python-libxml2  unity-lens-gwibber  unity-scope-video-remote   Finally, nothing I can do for this problem. And several days later, the root directory is full and I even could not uninstall any software because of this problem. What is the optimal path between $2$ fixed points around an invisible obstructing wall? Why my DAC works without GND and VCC? This is the most interesting thing I have seen! I have built a  that sends data (D0-D7) to a DAC (ADV2175). The design worked fine and I just have a low frequency alias noise (). This noise was present on all my ground planes (This is in prototyping stage and that's not unexpected from the test PCB).  I started tracing the noise path by disconnecting wires to see when the noise is gone. Surprisingly even after disconnecting all VCC and GND connections (+ all VREF and logics gone to 0 or 1) the noise was present yet. The more unexpected behavior is when the DAC is just connected to DATA port+ CLK pin (all coming from the source FPGA, NO VCC or GND), it continues working and a sine wave is produced (but with the same noise)!!!  First I thought it may receive its VCC/GND form the logic 1/0 fed into it but if it was correct, it should not have produce 0x00 and 0xFF levels, but it makes those levels correctly (I changed 0xFF data stage to another level and a notch appeared on top of the waves. this shows it can build oxFF correctly).    This is a big challenge for my PCB design, as if this device really takes its GND from the data port, instead of the dedicated pins, I will have a hard time for designing a reliable ground plane for it (the return voltage should go back to a logic noisy environment and analog-digital ground plane isolation may be impossible).  Can anyone explain what is happening there? Copy the contents of a file into the clipboard without displaying its contents How to copy the contents of a file in UNIX without displaying the file contents. I don't want to cat or vi to see the contents.  I want to copy them to clipboard so that I can paste it back on my windows notepad.  I can't copy the file from that server to another due to access restrictions. How can a computer with dynamic IP work as a webserver? How can a computer with dynamic IP work as a WebServer? How can I make a plane's edges taper inwards? How can I deform one edge or multiple edges of a plane or cube so they are tapered in and rounded equally? How do I ensure that a Python while-loop takes a particular amount of time to run? Confused By Copyright Rules I'm trying to understand this proof and I can follow it up the the second last paragraph where it states what happens if n is odd/even. I don't understand why there is just one conjugacy class when n is odd and two when n is even.    Help much appreciated, thanks! plist or sqlite