A Direct Summand of Finite-Free Module over a Local Ring is Again Free.
Finitely generated projective module
So far, I know if $p$ is a rational prime, then $(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$. $(2)$ If $p\equiv1\mod4$  then $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then $π_1 $ and $π_2$ are primes in $\mathbb{Z}[i]$. $(3)$ $2=(1+i)(1-i)$, then $(1+i)$and$(1-i)$ are primes in $\mathbb{Z}[i]$.  What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$? For example, $-3$ are prime in $\mathbb{Z}[i]$, but not in the above $3$ cases. Please note: I posted this question about an hour ago on my account (I just made it as I am new to the site). I revisited my post and saw that I had not given definitions of what I was asking (I thought they were common notations) and it would not let me edit the post for some reason so I will ask here now and hopefully someone can delete the other thread.  How to prove for a finite group $G$ that $|\langle g \rangle|=o(g)$?  Here are the definitions:  Let $G$ be a finite group and $g \in G$  $o(g)$ is the smallest $m \in \Bbb{N}$ such that $g^m=e$ (the identity element of the group). (If such an $m$ exists)  $\langle g \rangle=\{g^m : m \in \mathbb{Z}\}$  I really don't know how to show that this is true I tried to say let $o(g)=m$ and then show that $|\langle g \rangle|$ has exactly $m$ elements but I wasn't even sure it did because it would contain the elements $g^0,...g^{m-1}$ so that would mean it's true but what about $g^{-1}$ and the other elements with negative coefficients so it seems like it would actually be bigger than $m$?  Thanks. INT(5) vs SMALLINT(5): numbers in parenthesis after the numeric type In MySQL table definitions is there a different between INT(5) and SMALLINT(5)?  Or do they both represent the same size? Would Gollum qualify, as a Ringbearer, to go to the Undying Lands? Why do my xmodmap binds involving AltGr only work on some keys? I want to remap my AltGR+Right_Windows but I can't figure out how. No matter what I try, pressing the right windows key emits the name keysym when right alt is pressed or not.  These are the contents of the xmodmap file I am testing right now. In my keyboard, 38 is the keycode for "a", 48 is the keycode for aportrophe/doublequote and 134 is the keycode for the right windows key.  keycode  38 = 1 2 3 4 5 6 7 8 keycode  48 = 1 2 3 4 5 6 7 8 keycode 134 = 1 2 3 4 5 6 7 8   When I press Right_Alt+apostrophe, it outputs 5 but Right_Alt+a and Right_Alt+Right_Windows give 1 (both when right alt is pressed and when it isn't) which is not what I wanted.  My current keyboard layout is us(alt-intl) and the right alt is apparently set up to act as AltGr / Iso_Level3_Shift. The output of xmodmap -pm is the following:  xmodmap:  up to 4 keys per modifier, (keycodes in parentheses):  shift       Shift_L (0x32),  Shift_R (0x3e) lock        Caps_Lock (0x42) control     Control_L (0x25),  Control_R (0x69) mod1        Alt_L (0x40),  Meta_L (0xcd) mod2        Num_Lock (0x4d) mod3       mod4        Super_L (0x85),  Super_R (0x86),  Super_L (0xce),  Hyper_L (0xcf) mod5        ISO_Level3_Shift (0x5c),  Mode_switch (0xcb) So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ?  In other words, can a single parameter have jointly sufficient statistics? Postfix SMTP banner show multiple hostnames Why is electric field of an infinite plate constant at all points? I know from Gauss law, it is $\vec{E}=\dfrac{\sigma}{2 \epsilon_0}(\hat{n})$ at all points. But it doesn't make sense because of the inverse square nature of electric field which suggests if you move further away from the plane, electric field must reduce.   Then why is electric field of an infinite plate constant at all points? I have created a VF page and used "render as pdf"  &lt;apex:commandButton value="Save to Opportunity" action="{!savePdf}"  /&gt;   Its working fine when i am using above line without rerender and oncomplete but when i am adding rerender or oncomplete it is not working.I am able to save the PDF in attachment folder but when i am trying to open its showing  "Unable to load document"  Even if i remove oncomplete and use Rerender then also its saving the attachment in Attachment object and when i view ,it is not loading and says "Unable to load document"      &lt;apex:commandButton value="Save to Opportunity" action="{!savePdf}"  oncomplete="myClose();"/&gt;   Javascript Code  function myClose(){      self.close();     window.opener.location.href="/{!$CurrentPage.parameters.Id}";  }   Class Code  public void savePdf() {      PageReference pdf = Page.GenerateQuotePDF;     pdf.getParameters().put('id',parentId);      Attachment attach = new Attachment();      Blob body;     body = pdf.getContent();     attach.Body = body;     attach.Name = 'Opportunity'+ '.pdf';       attach.IsPrivate = false;     attach.ParentId = parentId;     attach.ContentType = 'application/pdf';     insert attach;     }     Please help.. Remove space before punctuation I need to remove locally spaces arround (especially before) punctuation signs. This spaces are added by french polyglossia.  \documentclass{article} \usepackage{fontspec} \usepackage{polyglossia} \setmainlanguage{french}  \newcommand{\nospace}[1]{\texttt{#1}}  \begin{document} \nospace{a:b} \end{document}     Thanks for help. Can the program counter on Intel CPUs can be read directly (that is without 'tricks') in kernel mode or some other mode? What do you do in Dungeon World when someone wants to perform an action they don't have a move for? So in Dungeon World, everything breaks down to 'moves'.  What about situations where a move is not available, but a check of some sort is called for, such as in the case of a non-fighter who wants to kick down the door?  What mechanism do you use to resolve that situation? Is there the equivalent of a skill check? How to create and design hoops net and attach ot to a hoop? How to iterate over arguments in a Bash script I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. As my client software uses lat/lon coordinates when communicating with my (spherical mercator) postgis database i decided to ST_Transform every geometry to WGS-84.  However i noticed that ST_Distance for WGS-84 returns units as degrees (i need meters). So i decided to use ST_DistanceSpheroid(geom1, geom2, 'SPHEROID["WGS 84",6378137,298.257223563]') but this method seems to be extremely slow.  Therefor i switched back to using spherical mercator again and transforming my input and output to and from wgs-84 as that performs a lot better.  Am i using the correct method or is this a known issue? How do shock waves form in space? Today my aerodynamics professor mentioned that the equations we were learning for atmospheric shock waves can also be applied by astrophysicists to study black hole-related shock waves in space.  How is this possible? Isn't space too rarefied for shock waves? What is an example application? He mentioned something about the density limit before/after a relativistic shock approaching 6. Does a single photon's energy depend on frequency? (photoelectric effect) Word for person unwilling to share information What do you call a person who is unwilling to share information? SSH login with clear text password as a parameter? I need to login to a user that I've created on a remote host running Ubuntu. I can't use an ssh key because the ssh login will happen from a bash script ran within a server that I won't have access to (think continuous integration server like Bamboo).   I understand this isn't an ideal practice, but I want to either set the remote host to not ask for the password or be able to login with something like ssh --passsword foobar user@host, kind of like MySQL allows you to do for logins.   I'm not finding this in man ssh and  I'm open to any alternatives to getting around this issue. Setting Access-Control-Allow-Origin in ASP.Net MVC - simplest possible method