Given $R$ a commutative ring and $I,J$ ideals of $R$, such that $I+J=R$, show that $I^m+J^m=R, \forall m \geq 1$   My problem is that I don't know what is the meaning of $I^m$ and in literature I didn't find anything. I know that, given $A,B$ ideals, then $AB= \{ \sum_{i=1}^n a_ib_i |a_i \in A, b_i \in B, n \in \mathbb{N} \}$, but with this notation $I^2=II=I$, so what is the meaning of $I^m$? It could be $I^m = \{ x^m | x \in I \} $?
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?
how many conditional sentence types are possible? I wonder how many conditional sentence types are possible. Basically, there are four main and two mixed types. By mixing tenses, it could be up to nine of them. Are the rest viable?  I came across the following examples in Full Spate Pre-Profficiency Primer:   After you have just sold alcohol to a young guy. If he is if under eighteen, you should not have sold him the alcohol. If he promised to bring the drinks, he will. If you will take a seat, the doctor will see you shortly. (Likely, the former will demonstrates politeness only.) I would be grateful if you would tell me a little about your friends. (The same politeness, but would instead.) If he will keep eating all those biscuits, he can not expect to stay as slim as he used to be.   I would be grateful if someone could shed more light on the topic. How can I convert my Java program to an .exe file? Is there a way to set my current location manually on Windows 8? Is there a way to set my current location manually on Windows 8? I don't have a GPS module, but the device is always located at the same place. The automatic detection via network sets a place 200 kilometers from here.  I searched for a setting that allows me to set the location by hand, but I didn't find one. I had some NEF files in a folder that I didn't do anything with. When importing them in Lightroom, no previews and a "file unsupported or damaged message" So I tried with Nikons ViewNX, nothing. Even not in DxO or PhotoMechanic.  I had a look with EXIFTool and the corrupt file prints me only this:  ExifTool Version Number         : 8.39 File Name                       : _DSC5559.NEF Directory                       : /Users/tunafish/Desktop File Size                       : 12 MB File Modification Date/Time     : 2010:11:29 12:46:56+01:00 File Permissions                : rwxrwxrwx Error                           : File format error   I saw another question on photo.stack  and tried to copy over the makernotes from a good file (_DSC5556.NEF) with  exiftool -tagsfromfile _DSC5556.NEF -makernotes _DSC5559.NEF   but EXIFTool gives me this error  Error: Not a valid TIFF - _DSC5559.NEF 0 image files updated 1 files weren't updated due to errors   I have uploaded 2 NEF's, with _DSC5559.NEF being the corrupt one here: Is it ever acceptable to have a  in your C or C++ application?  What if you allocate some memory and use it until the very last line of code in your application (for example, a global object's destructor)? As long as the memory consumption doesn't grow over time, is it OK to trust the OS to free your memory for you when your application terminates (on Windows, Mac, and Linux)? Would you even consider this a real memory leak if the memory was being used continuously until it was freed by the OS.  What if a third party library forced this situation on you? Would refuse to use that third party library no matter how great it otherwise might be?  I only see one practical disadvantage, and that is that these benign leaks will show up with memory leak detection tools as false positives. How to keep the console window open in Visual C++? I'm starting out in Visual C++ and I'd like to know how to keep the console window.  For instance this would be a typical "hello world" application:   int _tmain(int argc, _TCHAR* argv[]) {     cout &lt;&lt; "Hello World";     return 0; }   What's the line I'm missing? When should I use "shall" versus "will"? In a bash script what does a dot followed by a space and then a path mean? I came across this example when trying to mount a usb device inside a openvz container and I have never seen the construct in the second line before. Can you explain what it signifies?  #!/bin/bash . /etc/vz/vz.conf When will there be a Windows Phone app? What are the fault tolerances on the FDA food handling guidelines? Sometimes I have suggested that food that sat out a little longer than optimal may be safe to eat.    Granted, this is based on my experience, and I always qualify that it shouldn't be served to guests (and by extension, customers) and that one should determine their own tolerance to risk, but I want to know if there is information as to the methodology of the FDA guidelines.  Specifically, what is the contamination rate at n hours, and what are the initial conditions of the food and environment?  How much risk is there REALLY (in incidences per n-population and such) in a standard kitchen for food that has sat out 50%, 100%, 200%, etc. longer than recommended? Find conditional expectation from a 3-dimensional random vector? A good reference on optimal control theory Ok, so I am reading about decision making and I came across this subject. Fortunately it has a  , but the point is I want to see some examples, and learn to solve regular problems of this field. Of course I can go for the book by Pontryagin but then I don't think that would be an educative book. Does anyone has a suggestion? I would be happy if it also covers LQ problems. P.S. I studied math, therefore I appreciate a book with rigorous approach to the subject. I have a "term reference" field, in a form that uses an auto-complete widget. I want to limit user input to terms in the vocabulary only, and output a message when a user tries to enter a non existing term.  I found the  module. However this does not work with forms that contain a field collection or a field group.  There is also a contributed patch . But I was wondering whether there was a simpler/cleaner solution like a module or maybe I missed an already existing way of doing it with latest drupal and taxonomy versions. "There are several reasons proposed for the collapse of the bridge." There are several reasons proposed for the collapse of the bridge.   Is this present simple tense or the present perfect tense? I thought it might be the latter since there is a retrospective aspect to the sentence. However, I am quite unsure. Just freeze an application (suspend the CPU, save its state to disk) and 'defrost' it when I want to use it 4 weeks later. Command run in terminal, not with shortcuts As explain in this , I want to add shortcuts to switch workspaces (in Mate).  I found a way to switch workspace for the terminal: wmctrl -s $(wmctrl -d | grep "*" | awk '{print $1+1}').  But this command line does not work when I add it in the shortcut manager. Does any one know why ? What word or phrase best communicates the idea that a human (or other sentient being) is easily swayed, particularly between moral good and evil? Words I have found that are close are:   Impressionable - Easily influenced because of a lack of critical ability.   Implies that the subject lacks critical skills, and misses the concept of good and evil.  Gullible - Easily persuaded to believe something   Misses the concept of good and evil    I created a typical angels and demons scenario to demonstrate the polar extremes of the the spectrum. In this scenario, I lack the word to describe the humans, who can be influenced towards either end of the spectrum if impressed upon by an external entity.     The angels are devoted to good, and are immovable in their ideals. The demons are wicked, and will feign morality only to manipulate others. Both prey upon the humans, who are ____________.   I think that the key here is the spectrum of good an evil. Here are examples of other spectra and their associated words, for comparison.   Easily pushed from calm to angry: Temperamental Easily moved from determined to hopeless: Weak-willed Easily influenced by propaganda: Is there is any reason to make the permissions on an overridden C++ virtual function different from the base class?  Is there any danger in doing so?  For example:  class base {     public:         virtual int foo(double) = 0; }  class child : public base {     private:         virtual int foo(double); }   The  says that it is a bad idea, but doesn't say why.  I have seen this idiom in some code and I believe that the author was attempting to make the class final, based on an assumption that it is not possible to override a private member function.  However,  shows an example of overriding private functions.  Of course  recommends against doing so.  My concrete questions:   Is there any technical problem with using a different permission for virtual methods in derived classes vs base class? Is there any legitimate reason to do so? Limit values of struct member I have an android phone to which i have plugged an earphones. So at the top of the phone, I get the headphone symbol which indicates that the earphone is connected (In other words, the circuit at the 3.5 mm jack is closed).   Then I cut the two earphones (transducers) from it, and still the headphone symbol shows. When I later cut this cable, below where it branches out, even then it shows circuit completion.   So my question is this:   How does the phone detect circuit completion at the 3.5 mm jack and thus trigger all sound and music to be directed through the 3.5mm jack?