exercise: let $A$ be an integral domain in which all prime ideals are maximal (but may not be Dedekind in general).  let $P_1,  P_2$ be 2 distinct prime ideals.  prove: for any positive integer $m, n$,  $P_1^m+P_2^n=A$.   so the hint is to prove this first for arbitrary $m$ when $n=1$.  I finished this step by observing that if $P_1^m$ is contained in $P_2$,  then since they re prime ideals $P_1 \subset P_2$.  this contradicts the fact that they re distinguish and maximal.  so this was done.   but next how to use induction for $m$? I need some hint, thanks!
Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal?
Why does Calculate button remain gray in Group Stats Plugin for QGIS? In my case I failed to use  plugin (2.0.30) with QGIS 2.14.11 and 2.18.3 on windows64.  Tested on several datasets: points, polygons, and csv.- But I am not sure what I have missed.    I want to click on this Calculate button. Does there exist a laser that is tunable over the whole visible spectrum? Does there exist a laser that is tunable over the whole visible spectrum? If so does there exist one that is solid state? This is to settle a score with my dumb workmates. Is there an easy way to convert a number to a word in PHP? What order do triggered abilities happen in if multiple things trigger at the same time? I had several humans on the battlefield.  On turn three, I played . On turn four, I played . On turn five, during my upkeep, this question came up.  At this point, I need to draw/reveal three cards. If I reveal a human, Descendants' Path will allow me to cast it without paying its mana cost.  So, does it matter when I draw the human?  The static abilities state "at the beginning up your upkeep..." which leads me to believe that those two draws/reveals would happen before my normal draw. If that's the case, does the draw order happen in the same order that I played the enchantments?  In the end, that's how we decided to do it. I drew first for Triumph of Ferocity, then I revealed for Descendants' Path, then I drew for my turn. Did we do it right? What happens to spirits/souls of other creatues than Elves and Men after their death? I'm unable to access the following website from my Internet Explorer 8 under Windows XP with Internet Explorer unable to display web page error. By the other hand the FireFox browser at the same Windows XP can access this website.  How to solve the issue with IE8 ? How can I further diagnose my central air conditioning compressor fan motor and capacitor? I have a 15 year old Bryant Central Air unit in my house.  Yesterday it stopped working.  Here is what I have checked.   The breaker in the basement is not tripped. The furnace works fine from the thermostat. The blower downstairs by the furnace runs when AC is on and blows room temp air into the house, but the fan on the compressor is not spinning.  The fan on the compressor does spin freely when I manually spin it with a stick.  Checked for fuses in the outside box.  There are no fuses outside, only an on/off pull thingy with two metal plates that you have to turn upside down to turn off. The capacitor is not flat on top, but far less rounded then any of the examples of blown AC dual capacitors I've seen on youtube. I get 220v in on the contactor. I get 220v out on the contactor. I get 24v from the thermostat. When the air is on at the thermostat the contactor solenoid is in, when the air is off it is out.   When the air is on at the thermostat I hear a buzz coming from the outside compressor but I can't tell what the source of the buzz is. If I push the solenoid in when the AC is off, the compressor does not start up and the fan does not spin, I just hear the buzz. If I wait by the compressor outside for a few seconds while the AC is on there's another buzz sound that goes off for about 3 seconds then stops.  From listening to the AC when it was working this is the buzz sound that goes off before the compressor and fan usually kick in. After leaving the non-functioning unit on for a few minutes while I diagnosed I noticed that the metal on top of the compressor fan motor was VERY hot to the touch.  It cooled after I turned the AC off.   I'm thinking the fan motor on the compressor is dead or the capacitor is bad.  Is there a way to check the capacitor and the fan motor while they're still in the unit?  With the power off of course.  Does anyone think it's something else?  Do I have to buy the overpriced ($485 for the manufacturer fan motor and $200 for the dual capacitor) replacement part from the manufacturer or can I use a third party manufacturer?  Any other help would be appreciated.  As much as I'd love to bring an HVAC guy in here I just can't afford it.  Either I go without AC, or I fix it myself.  EDIT:  With the AC on I manually spun up the fan on the compressor and the fan started spinning and the AC unit fired up. Yesterday, I have downloaded and installed Windows 8 Professional, after downloading it (legally) and burning it to a DVD, on my 3-year-old  (Intel Core Duo 2x2.4&nbsp;GHz, 4&nbsp;GB RAM,  hard disk drive).   The installation failed two or three times, each time differently (error codes, for example 0xc000021a, with an infinite boot loop), but finally I completely formatted the hard drive and installed. Seemingly, everything was OK.  However, after booting successfully, the OS is extremely slow. When I open the task manager, it shows that the HDD is at 100% - however, no application or background process or Windows process can be made responsible. I searched the Internet and found that Windows might try to "index" the hard drive, but I did't find a process that isn't also on my second computer, which still uses the Release Preview (restarting every two hours). Also, the startup is empty, and HD Tune and HDD Health show that the drive health is OK, even though I don't know how reliable these two programs are.  I looked for diagnose software from Toshiba, but they only offer tools for "Fujitsu branded Toshiba devices"; I tried it out, and it did not work on my hard drive.  Does this sound like a hard drive failure? The laptop is 3.5-years-old, has been carried around sometimes roughly and was used (and on) most of the time. However, it would have failed exactly the night before installing Windows&nbsp;8 Profesional, which seems - unlikely.  If this is a software error - then what did I do wrong if this occurs from the moment of installation (even the personalization at the end of installation took half an hour)? How can I fix this? How do I create a key binding that allows me to mount my google drive? I want to mount/unmount my google drive using with a key binding without having to open a terminal. How can I set this up? Is there any research or data on horizontal vs. vertical navigational menus? I've seen many sites using both horizontal (top) navigation, vertical (left) navigation, and both across web sites for a long time. My understanding is that they're both conventions, but I would assume there must be some data out there supporting one route over the other.   My question is: are any studies or data for or against horizontal or vertical navigation (as suggested above) with regards to primary navigation (specifically on a smaller site or web app).  I've read some interesting things that are flipping me back and forth between liking horizontal more than vertical and vice versa. For example if you look at Jacob Nielsen's post on  you get the feeling that more important content should be on the left -- but you could take that as it being more important for the navigation (the user's escape routes) to be in that valuable space. Why is it possible to conclude everything from a false statement? Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) Allow users to hide connections between accounts This feature request is inspired by the question on Programmers.SE on .   Stack Overflow is a site aimed at professional programmers, and it is used by some employers and some employees to examine or showcase the candidates abilities. I don't see any problem with that, but there is one aspect I'm a bit concerned about. Certain SE sites are far more private than the professional sites like SO. Anyone that participates in Christianity, Judaism, Islam or Biblical Hermeneutics is likely giving out a significant amount of information about their personal belief or world view to anyone who can connect the user name to the actual person. Another example would be controversial topics on Skeptics or Politics; not everyone would like to give their employer access to that kind of information.  The fundamental problem is that the SE platform does not allow for any separation inside one account. Of course one could create separate accounts for private and professional topics, but that requires a significant amount of clairvoyance and a rather annoying user experience juggling separate browsers to log into different accounts at once. An additional problem is that SE has automatic user merge features that can be triggered by logging into two accounts in the same session. I'm not sure how they work exactly, but I suspect that having your two accounts accidentally merged is a real danger if you are not careful.  There should be a mechanism to hide the connection between SE accounts, without having to use separate OpenIDs. For example, if I would like to use my SO account in job applications, I should be able to separate it from all my other accounts. The separation would only be for the publicly accessible part, moderators could still be allowed to see the connection between accounts. And behind the scenes it would just work as it is now, just without any links between the accounts.   The aspects that would need to be changed that I can think of are   the accounts tab in the user profile, that shouldn't show any connections for separated profiles copying the user profile to other sites should not work for separated profiles the network profile should not show the connections between separated profiles Area 51 should also not show the profile connections   This separation would not have to be bulletproof, but it should prevent users googling a specific person to make connections the user does not want to make public. Why can't the "Launcher" Icons be made smaller? At the smallest setting, they are still 4 times bigger than they need to be on a 10" Netbook. The Icons in the "Top Panel" are much smaller and we users don't have a problem selecting them. Google Chrome on a fresh CentOS 7.1 not starting up Cut one side of a rectangle node in TikZ I would like to draw two consecutive rectangle nodes (one is next to the other with an intersected edge). How to cut one side of a rectangle node in TikZ so that the intersected side looks like others?   Consider the example below, I don't want the "duplicated" edge between node 1 and 2.   \begin{tikzpicture}    \node [style={draw, inner sep=0pt, minimum size=0.5cm}]           (1) at (0, 0)  { $1$ };   \node [style={draw, inner sep=0pt, minimum size=0.5cm}]           (2) at (0.5, 0)  { $2$ };  \end{tikzpicture} How to get Url Hash (#) from server side I know on client side (javascript) you can use windows.location.hash but could not find anyway to access from the server side. I'm using asp.net. Is there a cross-browser onload event when clicking the back button? For all major browsers (except IE), the JavaScript onload event doesn’t fire when the page loads as a result of a back button operation — it only fires when the page is first loaded.  Can someone point me at some sample cross-browser code (Firefox, Opera, Safari, IE, …) that solves this problem? I’m familiar with Firefox’s pageshow event but unfortunately neither Opera nor Safari implement this. $L^p$ and $L^q$ space inclusion Let $(X, \mathcal B, m)$ be a measure space.  For $1 \leq p &lt; q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied? Share wireless Internet connection through ethernet What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)?  It seems that there is no clear reason to prefer over other axioms..  Thanks for help.