I am trying to prove the following result:     If $R$ is a commutative ring with identity, $I$ is an ideal of $R$ such that $I^2=\{0\}$ and $a+I$ is an idempotent element in $R/I$ then there are some idempotent element of $R$ in the coset $a+I$.   What I tried to do?  Well, suppose that $a+I$ is idempotent in $R/I$, then $(a+I)(a+I)=a+I$, so, $a^2+I=a+I$ and then $a^2-a \in I$. We know also that $1-a+I$ is idempotent since $(1-a+I)(1-a+I)=1-2a+a^2+I = 1-2a+a+I=1-a+I$. And then, from this we know that $a(1-a)+I$ is idempotent.  We need to show that there exist some $m \in I$ such that $(a+m)^2=a+m$, but $(a+m)^2=a^2+2am+m^2=a^2+2am$ since $m^2=0$, so $m$ needs to satysfy $a^2+(2m-1)a-m=0.$  I dont know if this reasoning is right, neither if this is a way to go over it. Can someone give me a hint to finish it?
The problem is this:      Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent.   I have spent a few hours rolling around in abstracta with no destination. I believe that if I could write down a concrete example of this, the example could guide me through the abstract definitions and show me where to look for an idempotent in $R$. I have looked at 2x2 matrices and could not find any such examples. Might anybody have one?
Evaluate the following limit of finite sum Evaluate the limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=0}^{[n/2]}\cos \left(\frac{k\pi}{n}\right)$$  I tried through considering two cases : (i) When $n$ is even (ii) when $n$ is odd.  When $n$ is even then the limiting value is $\frac{1}{\pi}$. But when $n$ is odd then ??  Case 1 : $n$ is even..  Then , take $n=2p$ where $p$ is positive integer. Then $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=0}^{[n/2]}\cos \left(\frac{k\pi}{n}\right)$$  $$=\lim_{p\to \infty}\frac{1}{2p}\sum_{k=0}^{p}\cos\left(\frac{k\pi}{2p}\right)$$  $$=\frac{1}{2}\int_0^1\cos(\pi x/2)\,dx$$  $$=\frac{1}{\pi}.$$  Case 2 : When $n$ is odd.  Take, $n=2p+1$ , where $p$ is a positive integer. Then ,   $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=0}^{[n/2]}\cos \left(\frac{k\pi}{n}\right)$$  $$=\lim_{p\to \infty}\frac{1}{2p+1}\sum_{k=0}^p\cos\left(\frac{k\pi}{2p+1}\right)$$  From here how I proceed ? Build a basic Python iterator I currently have a masters degree in an engineering field and I don't currently have any PhD degrees (I may study for a doctorate degree in future). It happens that when I go to conferences or scientific occasions, even in the emails I receive from people; they use a doctor prefix for my name; but I do not have a PhD degree.  What is a good etiquette to deal with such thing? Should I directly tell them that I do not have a doctorate degree? Or should I neglect such a mistake by them and don't do anything?  I think it is unethical not to correct such thing, because the person may still think that I have a PhD degree; and such wrong usage of title should be corrected some way. Once a person used doctor title for my name in an email and I just added a line in my reply to his email, stating that I have a masters degree. I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$ Travel in schengen countries without carrying passport for non EU citizen How do I create a permanent Bash alias? I would like to create an alias to rm command in order to have a confirmation message after executing this command. So I am creating an alias like this alias rm='rm -i'. But as far as I know this is a temporary alias and it lives until you close the terminal.  As it is explained  to save alias permanently I need to execute ~/.bash_aliases or ~/.bashrc commands in terminal and add my alias there. But when I execute ~/.bashrc I get following error message :  bash: /home/bakhtiyor/.bashrc: Permission denied   When I run ~/.bash_aliases I get another error message like this:  bash: /home/bakhtiyor/.bash_aliases: File or directory doesn't exist.   What is the actual problem and how can I solve it? override at runtime __setattr__ How do I  prevent .xsession-errors from eating disk space? Is there any way to completely disable the .xsession-errors file?  I have it as a symlink to /tmp so that my laptop hard disk can (hopefully) go to sleep for once, but at least 95% of the stuff in the file (it fills up at about 500k an hour) is...  (nautilus:1618): GLib-GObject-CRITICAL **: g_value_get_object: assertion `G_VALUE_HOLDS_OBJECT (value)' failed  which is total garbage to me.  I have tried doing a symlink to /dev/null but that does not work (the resulting link is overwritten), and I also do not want the activity so that hopefully my laptop can go to sleep for once.  I am using Ubuntu 11.04, with no special add-ons to Nautilus. latex gives me the error: Ouch---my internal constants have been clobbered As the title states, compiling with latex produces the error, right away:  Ouch---my internal constants have been clobbered!---case 14   I have no idea what this means... I use TexLive 2012. I updated my texmf.cnf and ran fmtutil-sys and texhash. I guess texhash may have messed it up?  Any ideas?  As requested a minimal working example:  \documentclass{article} \begin{document}     Any tex file. \end{document} I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? Why do I get the error "failed unmounting /var" during shutdown? I have a fresh install of Debian 9.0.0. It has a separate 1 GB partition for /var formatted at ext4. When I shutdown from the terminal, using shutdown -h now, there is a bright-red error message stating "failed unmounting /var".  What is the fix for this error? Search/browse my own questions easily Why the middle mouse button in Blender 2.80 changes the orthographic view to perspective? If I choose an orthographic view by using the numpad and then try to orbit the geometry by clicking the middle mouse button Blender goes from orthographic to perspective. Why is that the default now? Wouldn't it be more logical if instead of that Blender kept the orthographic mode? In other words: Why Auto Perspective is turned on by default? How can I block app and game invitations once and for all in Facebook? I want to block app and game invitations once and for all in Facebook. How can I do it so they cannot disturb me anymore? /A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? Brown sugar instead of white sugar I have searched, but not found anything on this. I am looking for a functionality in bash, using a terminal.  Way back when, I had a user on a debian system, and a friend set me up with a convenient history search feature (I believe I used tcsh then), where I would type the beginning of a previous command, hit up-arrow, and it would do a search, based on the partial string.  E.g. if my history is:  ./script.pl ./script.pl arg1 cat output cat output | grep yada   And I type ., and press up-arrow, it would show me: ./script.pl arg1. Press it again and it would show ./script.pl, etc.  Very much like it would perform a grep on .bash_history. Is there a way to get this functionality? I am not talking about miracles, religious revelation, or artistic expression, but something more mundane. There is a lot of "empirical" evidence that the Riemann hypothesis is true, the scare quotes indicate that the source of the evidence is "mathematical intuition". In psychology there are mental states that no one sees, touches, hears, smells or tastes, qualia with their perceptual fullness that can not be conceptualized or directly communicated, and abilities/skills "know how" with the same problem. In linguistics Kripke's theory of reference relies on "modal intuitions" to decide the truth of counterfactuals. But an empirical counterfactual is an oxymoron, especially when it is embedded into an entire "possible world".  Freudian psychoanalysis and Husserlian phenomenology are attempts to deal with some of them systematically, but they are traditionally confined to the fringe of science at best. The mainstream approach is empirical second hand, but behaviorism in psychology and linguistics, and nominalism in mathematics, were not very productive.   According to the naturalized epistemology there is no a priori "first philosophy" of science, methodology is subject to the "tribunal of experience" and revision, just as the science itself. Should it apply to non-empirical experience? Sense empiricism has served science well for centuries, and was fully embraced by Quine.  takes him to task for it:     "Cognitive science is an empirical science working to unearth the mechanisms through which natural language constitutes itself. That account has had to recognize the indispensability of mental states, of beliefs, if it is ever to become adequate to the problem... There is still too much "first philosophy" in Quine. We must rescue naturalized epistemology from its own founder".    Note the use of "empirical" together with "mental states".  Questions: Does naturalized epistemology dictate that natural science should relax sense empiricism, and approach non-empirical phenomena partly first hand? If so, how can the scientific method be adapted to introspective/intuitive phenomena that are not readily reproducible, manipulable, measurable, and/or publicly accessible? What would play the selective role of empirical testing? Can there be non-empirical (more likely, not entirely empirical) natural science? Contextually grey out panel element in python 2.8 I have a panel with split column.  In the 2nd row is 3 boolean variables. I want to grey out the 2nd and 3rd option by default. I want the 2nd to be changeable if 1st is true, and 3rd to be changeable if both 1 and two are true. I've read something about "poll()" but not how to actually use it, or set conditions. Physics should only be enablable if singleMesh, and destructable only if physics enabled.      # Create two columns, by using a split layout.     split = layout.split()      # First column     col = split.column()     col.label(text="Building Size:")     col.prop(bpy.context.scene.user_vars, 'buildingHeight')     col.prop(bpy.context.scene.user_vars, 'buildingWidth')     col.prop(bpy.context.scene.user_vars, 'buildingDepth')      # Second column, aligned     col = split.column()     col.label(text="Dynamics:")     col.prop(bpy.context.scene.user_vars, 'singleMesh')     col.prop(bpy.context.scene.user_vars, 'enablePhysics')     col.prop(bpy.context.scene.user_vars, 'destructable_Building')  #EDIT This is what the specific solution to my example looks like. col = split.column() col.label(text="Dynamics:") col.prop(vars, 'singleMesh') sub = col.row() sub.enabled = vars.singleMesh sub.prop(vars, 'enablePhysics') sub = col.row() sub.enabled = vars.enablePhysics sub.prop(vars, 'destructable_Building') Today, on C.SE, I reviewed some 'Late Answers' and First Posts.  I was not even near my daily vote limit.  And yet, I was given a message that said      comments disabled on deleted / locked posts / reviews   It seems I was not the only one either:   The workaround, of course, was simply to go back and provide the guidance - but was this a design or a bug?  If it is design, why would you want to discourage the feedback? Especially with first posts, there is often a high percentage of 'didn't read the FAQ' usage, which can best be addressed by gently providing guidance. (e.g. "Unfortunately, this doesn't answer the question" or "This reads like a comment" or "This really isn't on-topic here - can I suggest you checkout our [faq]" etc...)  Timely first post guidance goes a long way towards making good users.   If it was a bug, was other functionality silently rolled out?