Proper ideals of local rings Let $R$ be a commutative ring with $1$. We call $R$ a local ring when $R$ has exactly one maximal ideal $m$. Let $I \subset R$ be a proper ideal of $R$, i.e. $I \neq R$.     Question: Is $I$ contained in $m$?   Approach: I think this is true. If $I$ is a proper ideal of $R$, then $I$ does not contain any units of $R$ (otherwise we would have $I=R$, contradiction!). But this is the only thing I can think of.  Could you please help me solving my question? Thank you!
Every proper ideal is contained in a maximal ideal, in a commutative ring with identity. The statement is:     In a commutative ring with 1, every proper ideal is contained in a maximal ideal.   and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset  A\mid A\text{ is an ideal}\} $, then by set inclusion, every totally ordered subset has a bound, then $P$ has a maximal element $M$.  My question is why $M$ must contain $I$?
Windows Server 2016 scheduled task schedule must be in future We have a 2012 server that has about 20 scheduled tasks that are used for monitoring.  We have built a 2016 server to replace it and as I move tasks to the new server I have run into an issue.  Most of our tasks run every few minutes.      In Server 2012 we would schedule the task to start at 1200AM of the current day (so a time in the past) and reoccur every X minutes.  The task would start at its next scheduled time.  Perfect.  In Server 2016 schedules that start in the past don't run.  So we had to schedule it to start in the future.  This is fine until you reboot.  Now the start time is in the past and it won't run.  It doesn't even register a failed run.  The Next Run Time column shows that it should run but doesn't.    Besides creating two triggers, one that starts in the future and another that starts on reboot (which we don't want to do since some tasks should only run at specified times), is there a better way?  This seems to be a bug.  The 2012 version worked great.  UPDATE  There is a workaround for this.  Create a one time schedule that runs indefinitely.  This works after restarts just like you would expect.  I know this isn't great for schedules that are "weird" but works fine for things that we want to run every 3 minutes or whatever. Write18: capturing shell (script) output as command/variable? Let's say I compile the following file with pdflatex -shell-escape test.tex:   \documentclass{minimal}  \begin{document}  File listing is:  \immediate\write18{ls /usr}  \end{document}   This will send the output of the command ls /usr to report/log of pdflatex (primarily to stdout).   There are then two cases I'd like to utilize:    The output of ls /usr being included directly in the document (LaTeX stream). The output of ls /usr becoming the contents of a \newcommand (by this I mean, that I'd like the script to be executed when the \newcommand is executed first - and upon subsequent calls to the new command, the shell code should not be executed anew).    I have read through , but I'm not sure if this "pipe input" can be applied to \newcommand.   I have also read through  - and it seems that Tex' file I/O should be used; but I'm quite disliking the fact that I'd still have to redirect the script output (actually, in this case, the ls /usr output) to a file, and then read it in, to have it as contents of a command.   So, is there an easier way to achieve what I'd want (hopefully, illustrated through an example based on the above code)?     EDIT: Ehm, I should have asked one more question earlier :) I'll try with an edit here, although it will probably get missed .. :)   I originally asked for a \newcommand that will execute shell code only upon its definition (i.e., in a sense it is "cached"); and the answer from @egreg does exactly that. But then - would it be possible to have a different \newcommand definition, such that each time this newcommand is called, the shell command is executed anew? I.e. executing something like \@@input|"cat tempfile" (which cannot be executed as such), where tempfile changes between calls? If $\cot A+\cot B+\cot C=\sqrt3$ then prove that the triangle is equilateral triangle.   Trial: I can counter check that this is true as $\cot 60+\cot 60+\cot 60=3 \frac{1}{\sqrt3}=\sqrt3$.Here I also know that $A+B+C=180$. But I am unable to show it. Please help. How to tell the difference between linear and non-linear regression models? Is a passport needed when flying within Mexico as a U.S permanent resident? This one-liner removes duplicate lines from text input without pre-sorting.  For example:  $ cat &gt;f q w e w r $ awk '!a[$0]++' &lt;f q w e r $    The original code I have found on the internets read:  awk '!_[$0]++'  This was even more perplexing to me as I took _ to have a special meaning in awk, like in Perl, but it turned out to be just a name of an array.  Now, I understand the logic behind the one-liner:  each input line is used as a key in a hash array, thus, upon completion, the hash contains unique lines in the order of arrival.  What I would like to learn is how exactly this notation is interpreted by awk. E.g. what the bang sign (!) means and the other elements of this code snippet.  How does it work? Theorem style with name as argument Is there some way to get a theorem type which will make a theorem with the name passed as a parameter? In other words, is there a way to define "named" such that   \begin{named}[NAME] text \end{named}   will print   "NAME: text"?  I tried using the definition \newtheorem*{named}{}, but that results in   "(NAME): text", which is quite different from what I am trying to do.  I know that I could also create a new instance of \newtheorem for each named theorem I want to use, but that is clunky and inconvenient. Is there a better way? Can I cast a spell while another spell or ability is resolving? My opponent has a  with scads of loyalty. I've got a  and a  on the battlefield.  He activates Sorin's "ultimate" ability, targeting Gideon.     -6: Destroy up to three target creatures and/or other planeswalkers. Return each card put into a graveyard this way to the battlefield under your control.   Is there any way I can use the Spellbomb to prevent my opponent from stealing Gideon after he is sent to my graveyard? Auto mount a disk drive in ubuntu 14.04 LTS (Bookmark directory not found) I'm using ubuntu 14.04 LTS. When I start my pc and click on any bookmark from the file manager it shows this error:   Unable to find the requested file. Please check the spelling and try again. Unhandled error message: Error when getting information for file '/media/angel/DOC/edu': No such file or directory   But when I open DOC drive and again clink on the same bookmark it works properly. Can anybody tell me why is this happening and how to solve this issue?   Edit: Output of lsblk is following: (ubuntu is installed in sda5)  NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 119.2G  0 disk  ├─sda1   8:1    0  91.3G  0 part  ├─sda2   8:2    0     1K  0 part  ├─sda5   8:5    0  24.1G  0 part / └─sda6   8:6    0   3.8G  0 part [SWAP] sdb      8:16   0 465.8G  0 disk  ├─sdb1   8:17   0  93.2G  0 part  ├─sdb2   8:18   0     1K  0 part  ├─sdb5   8:21   0  93.2G  0 part  ├─sdb6   8:22   0  89.3G  0 part  ├─sdb7   8:23   0  93.2G  0 part  ├─sdb8   8:24   0  93.2G  0 part  └─sdb9   8:25   0   3.8G  0 part  sr0     11:0    1  1024M  0 rom     After opening the disk drive the output of lsblk is bellow:   NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 119.2G  0 disk  ├─sda1   8:1    0  91.3G  0 part  ├─sda2   8:2    0     1K  0 part  ├─sda5   8:5    0  24.1G  0 part / └─sda6   8:6    0   3.8G  0 part [SWAP] sdb      8:16   0 465.8G  0 disk  ├─sdb1   8:17   0  93.2G  0 part  ├─sdb2   8:18   0     1K  0 part  ├─sdb5   8:21   0  93.2G  0 part  ├─sdb6   8:22   0  89.3G  0 part /media/angel/DOC ├─sdb7   8:23   0  93.2G  0 part  ├─sdb8   8:24   0  93.2G  0 part  └─sdb9   8:25   0   3.8G  0 part  sr0     11:0    1  1024M  0 rom   The output of sudo blkid gives the following information about sdb6:  /dev/sdb6: LABEL="DOC" UUID="04280A22280A12FA" TYPE="ntfs" Browsing the internet from the command line How to install KDE along Unity? I have just installed the latest Ubuntu 14.04 and I would like to install KDE along side my Gnome/Unity that I have now. How can I do this?  Thanks! Can I see images and watch movies inside the terminal emulator check if numbers from command line are powers of 2 BM01 2011/12 Question 6 Geometry Problem Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A,B$ and $C$ are $D, E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds. The altitude from A is the line through A which is perpendicular to BC. The foot of this altitude is the point D where it meets BC. The other altitudes are similarly defined.   Thanks in advance for any contributions. Can prematurely powering down a computer cause bad sectors in a hard drive? Q1: Can forcibly powering off a PC without using the built-in shutdown/reboot functions, cause a HDD to accumulate bad sectors? Whether it be the result of BSODs, lockups, crashes, etc.   Q2: If so, are these bad sectors repairable? I know an OS typically relocate bad sectors to other areas of the drive, but once a drive accumulates too many it just dies.   I have read up and saw things about 'soft' and 'hard' bad sectors. One caused by software and the other being actual physical damage. The reason I am asking this is because I have had several HDDs die on me in the past year, far before their estimated lifespan, and one thing they all had in common was that I had forcibly powered off my PC a lot. How do I set up an account with multiple users? Virtualbox Kernel driver not installed I tried to use my Virtual Box and this is the error I received - (I just updated to the new Ubuntu also) and I am VERY new at this Linux OS.     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or   there is a permission problem with /dev/vboxdrv. Please reinstall the   kernel module by executing      '/etc/init.d/vboxdrv setup'      as root. Users of Ubuntu, Fedora or Mandriva should install the DKMS   package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary. Working with a huge file. Illustrator is super slow. What can I do to ease the load? I just spent the last 10 minutes trying to put a new set of grips on to my mountain bike handlebars..  The old ones came off fairly agreeably and I've got the new ones 80-90% of the way on, however, at this point both my hands are stinging from the aggressive twisting and pushing - I have tried gloves, but I feel it's best left till my hands don't hurt anymore.  Is there a tried and tested way of doing this? Or is there a knack to this that I am missing? Handling multiple regex matches in Ampscript Given this string:  SET @text = "&lt;a href=""https://salesforce.stackexchange.com""target=""_blank""&gt;Salesforce&lt;/a&gt; &lt;a href=""https://stackoverflow.com"" target=""_blank""&gt;Everything else&lt;/a&gt;"   And this function:  SET @matching = '\s+(?:[^&gt;]*?\s+)?href="([^"]*)"' SET @output = RegExMatch(@text, @matching, 1)   How can I manage to find and print an arbitrary number of matches in my string? E.g. in above example, I would like to present e.g. a rowset with each URL in each row.