Let $R$ be a local ring (commutative with identity) and suppose we have $M$ and $N$ submodules of $R^n$ such that $M\oplus N\cong R^n$. Then $M\cong R^s$ and $N\cong R^t$ for some $s$ and $t$ such that $s+t=n$.   I am completely lost here.
Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free?    What I know about a finitely generated projective module $M$ over a PID $R$ is isomorphic to $R^k\oplus R/(a_1)\oplus\dots\oplus R/(a_n)$, and for the local ring case I don't know how to start.
Proof of continuity of Thomae Function at irrationals. If low-cost index funds are considered the best investment, why are there so many high-cost, managed funds? Why are the shebang parts of the following elementary scripts not working:  $ cat hello.sh #! /bin/sh echo Hello $ ./hello.sh bash: ./hello.sh: /bin/sh^M: bad interpreter: No such file or directory  $ cat hello.py #! /usr/bin/env python3 print("Hello") $ ./hello.py : No such file or directory   whereas calling the interpreter manually is working:  $ sh hello.sh Hello $ python3 hello.py Hello Creating point features with exact coordinates in QGIS? Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ Determine the set of points $z$ that satisfy the condition $|2z|&gt;|1+z^2|$  I tried to redo this problem and got to this point  $|2z|&gt;|1+z^2|$ $\Rightarrow$ $2|z|&gt;1+|z^2|$ $\Rightarrow$ $2|z|&gt;1+z\overline z$  Let $z=x+iy$ then   $$2\sqrt{x^2+y^2}&gt;1+(x+iy)(x-iy)$$  $$2\sqrt{x^2+y^2}&gt;1+(x^2+y^2)$$  $$0&gt;1-2\sqrt{x^2+y^2}+(x^2+y^2)$$  $$0&gt;(1-\sqrt{x^2+y^2})^2$$  since $x,y$ are real number , so there is no solution for this inequality? Detecting unused packages in LaTeX document I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? In France the higher education is like this:   3 years of "Licence" 2 years of "Master" 3 years of "Doctorat"   But what about in the United States?  I heard that they have things called "undergraduate", "graduate" and "postgraduate". How many years do each of those take? And what are the equivalencies between these and the "Licence - Master - Doctorat"?  Note: If the answer to this question depends on the field of studies, then the field is physics. Residue fields of schemes of finite type (over $\mathbb{Z}$) Suppose $X$ is a scheme of finite type over $\mathbb Z$. I want to prove that:  (1) The residue fields of closed points of $X$ are finite;  (2) For a given $q=p^n$ with $p$ prime, there is only a finite number of closed points of $X$ whose residue field is $\mathbb F_q$.  For (1), I see that one has to use some form of Nullstellensatz. First, we suppose $X=\operatorname{Spec}(A)$ affine, and $A=\mathbb{Z}[X_1,...,X_n]/I$. We need to show that if $m$ a maximal ideal of $A$, then $A/m$ is finite. Note $p=\mathbb Z\cap m$. If $p$ is maximal, we get that $A/m$ is of finite type over $\mathbb F_p$ thus a finite extension by Nullstellensatz and we are done. But $p$ could be the $(0)$ ideal in which case I neither conclude nor get a contradiction.  For (2), we can again argue locally, so we need to show that $A$ contains only a finite number of maximal ideals $m$ with $A/m$ of fixed cardinality $q=p^n$. I get an impression that to count these ideals its the same as to count the number of automorphisms of $\mathbb{F}_{q}$ which fix $\mathbb F_p$, but the corrspondence is not bijective, so we only get an upper bound which is enough. Is this idea correct? Simple request, really.  Can we add a detection for the     Please avoid extended discussions in   comments. Would you like to   automatically move this discussion to   chat?   prompt, so that if one of the users in the "extended discussion" has under 20 rep it doesn't appear?  I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them.  New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat. Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$ form in a .tpl.php file This is the code in my template.php file.  function studiga_theme_theme($existing, $type, $theme, $path) {         $items['add_property_form'] = array(         'render element' =&gt; 'form',         'template' =&gt; 'node--14',         'path' =&gt; drupal_get_path('theme', 'studiga_theme') . '/templates/node',     );      return $items; }    function custom_add_property_form($form, &amp;$form_state) {     $form['first_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('First name')),     );     $form['last_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Last name')),     );     $form['address'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Address')),     );     $form['city'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('City')),     );     $form['state'] = array(         '#type' =&gt; 'select',         '#options' =&gt; array(             'default' =&gt; 'State',             '...' =&gt; '...',         ),     );     $form['zip'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Zip')),     );     $form['email'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Email')),     );     $form['phone'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Phone')),     );     $form['submit'] = array(         '#type' =&gt; 'submit',         '#value' =&gt; 'Submit',     );      return $form; }   This is the code in node--14.tpl.php.                                          Contact Information                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  &lt;div class="row"&gt;     &lt;div class="small-12 medium-12 large-8 large-offset-2 columns"&gt;         &lt;?php print render($form['submit']); ?&gt;     &lt;/div&gt; &lt;/div&gt;     But no fields are appearing - what am I doing wrong? You can't have a straight forward  in a tpl.php file can you? As a new user, I find it impossible to contribute to the community or to even begin contributing. I know why the reputation system is in place and I respect it, but at the same time it is too hard on new users.  It started with me wanting to upvote a question...     "Vote Up requires 15 reputation"   Just to appreciate a good question that helped me I need 15 rep? OK..., so I find out how I can get reputation points. More or less, I need to ask a question for +5 and add an answer for +10. I don't have any questions at the moment and I don't want to spam a random question, but I did happen to have a slightly different answer to the earlier question that I wanted to upvote. So I try to answer...     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.   Now... I know this doesn't apply to every question, but at the same time, this means I'm forced to answer yet another question, and I don't want to go around looking for random questions and spam irrelevant answers...  Now I'm stuck. So I try to talk to some other users in the chat rooms about how to go about getting reputation points and how they first started. I entered into a "casual chat room".     You must have 20 reputation on Meta Stack Overflow to talk here.   Yes... I can understand why this is enforced, because we don't want new users to spam chat rooms, but all I wanted to do was ask for help. So I can't even talk to anyone about my newbie problems. So okay, I'll just settle for upvoting comments in the question that helped me. I don't see an upvote hover-over button so I can't even do that.    So now I'm stuck at 1 reputation point, unable to cross that 15 rep barrier, and even after posting this question I'll only have 6 rep points and still unable to do anything, and I'll be forced to look for questions to answer. Yes, I should be contributing to the community, but I shouldn't need to begin answering or making questions to upvote questions and answers. This only spurs more spam/irrelevant answers if anything.   Very, very, very new-user unfriendly. Why do we use the base form of verbs instead of the past form in past tense questions? In regard to this , my question is similar but that answer is not clear. I want to know why we use base form of verb, e.g. 'go' to form the past tense instead of past form such as 'went'?     Question: Did you go to a bank?  (Why we use 'go' here, rather than 'went'?)   Answer: I went to a bank.   Am I confusing verb forms with tenses? Please note that I'm not a native speaker so some clarification or reference on this would be helpful. Simple as the title states: Can you use only Java commands to take a screenshot and save it?  Or, do I need to use an OS specific program to take the screenshot and then grab it off the clipboard? How Did Dumbledore Stay in Limbo? Most of this discussion involves events at the end of Harry Potter and the Deathly Hallows so if you haven't read that book and don't want spoilers, then stop reading right now.  Yes, now.  It's not clear whether limbo is something everyone moves through in Harry Potter when they die, but when Harry is killed, he not only ends up there, but Dumbledore is there, waiting for him.  Did Dumbledore wait for him all that time, in limbo, until Harry showed up?  Did he have any way of watching what Harry was doing in that time?  Or did he have a way to enter limbo when Harry did to meet him there?  And once Harry was gone, was Dumbledore able to move on, or would he be stuck there? How do I do this? I created a user without a home directory and now I want to create one. Not just a folder called new_user, I want the default home directory with all it's folders and hidden files, etc. Desktop folders won't open with clean install of 18.04 Why doesn't a fly fall off the wall? What is the difference between .profile and .bash_profile and why don't I have a .profile file on my system? So I'm pretty new to Linux and recently installed Fedora 19 on my netbook. I want to have a deeper understanding of Linux and the CLI so I'm now reading .  In exercise 3 the author gives the following commands:   ls -al cat .profile   I try to follow these instructions but to no avail. There is no .profile file in my user name directory as I can see from ls -al. From what I've learned the .bash_profile and .profile file are practically the same. The only difference I concluded from my research is that .bash_profile gets checked first.   What I don't understand is why are there multiple files for the same function in Linux, why is the Author using .profile and not .bash_profile (if I understand correctly .profile won't be read when the system finds .bash_profile first) and how comes that I don't have the .profile file on my system? The question at hand is:  "Two capacitors of capacitances $C_1$ and $C_1$ have charge $Q_1$ and $Q_2$. How much energy, $\Delta w$, is dissipated when they are connected in parallel. Show explicitly that $\Delta w$ is non-negative."  I'm confused about what the physical situation is. I took the assumption that these capacitors were somehow pre-charged, and then connected to each other in parallel without a voltage source. However, I don't understand how this would work. If a circuit is composed of only 2 elements, I don't see how they could be in any arrangement but series. Nevertheless, I tried solving it that way:  Two capacitors in parallel have the same voltage drop. Charge will be redistributed to make it the same voltage for both. Let $Q_1'$ and $Q_2'$ be the charges on the capacitors after they are connected. Now, picture the equivalent capacitor  $C_{eq} = C_1 + C_2 =$ $\frac{Q_1' + Q_2'}{V_f}$   conservation of charge:  $Q_1' + Q_2'= Q_1 + Q_2$,        $C_{eq} =$ $\frac{Q_1 + Q_2}{V_f}$  $V_f = $$\frac{Q_1 + Q_2}{C_1 + C_2}$  The initial energy of the capactiors is:  $U_0 = $$\frac{Q_1^2}{2C_1}$$+\frac{Q_2^2}{2C_2}$  $U_f = $$\frac{1}{2}$$(C_1 + C_2)V_f^2 = $$\frac{(Q_1 + Q_2)^2}{2(C_1 + C_2)}$  $\Delta U$$ = $$\frac{(Q_1 + Q_2)^2}{2(C_1 + C_2)}$-$\frac{Q_1^2}{2C_1}$$-\frac{Q_2^2}{2C_2}$  However, I don't see how this is necessarily non-negative.