For any module $M$, the functor $-\otimes M $ is right exact. I want to show that if $F$ is a free module then $-\otimes F$ is a exact.   I could not see how to adapt the proof for the case of $M$ not being free to  this case.
Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks!
This is a  about DNS (Domain Name Service).   If my understanding of the DNS system is correct, the .com registry holds a table that maps domains (www.example.com) to DNS servers.   What is the advantage? Why not map directly to an IP address? If the only record that needs to change when I am configuring a DNS server to point to a different IP address, is located at the DNS server, why isn't the process instant? If the only reason for the delay are DNS caches, is it possible to bypass them, so I can see what is happening in real time? I have a collection of files and directories with strange names, they contain spaces and the special characters \, *, ), ( and ?. All of them cause problems when I try to use them in the command line or scripts, e.g.:    $ cd space dir bash: cd: space: No such file or directory  $ cat space file cat: space: No such file or directory cat: file: No such file or directory  $ cat ( bash: syntax error near unexpected token `newline'  $ echo content &gt;\ &gt; ^C  $ ls ? (  )  *  ?  \   How do I enter file or directory names that contain special characters in the terminal? Is it possible to import comments into a chat room without the link appearing? Question: For an answer on a stack exchange site with say 8 comments, how can I / is it possible to import the comments into a chat room? Background: I was helping with a  and made several comments before I saw the link to auto-import comments into a chat room.  This was awesome by the way!! Note that the person who asked the question left a link to the chat room in a comment. I wanted to make a similar chat room  because I edited an answer which wasn't mine and added a paragraph.  I wanted to chat with the answer's author @JonHopkins to say, 'hey feel free to remove my edit -- just wanted to add my $0.02' -- especially as the stack exchange engines said not to add an additional answer.  HOWEVER, there was no link to auto-import the comments -- probably because I hadn't previously left one, but I'm not sure. So I ask here, Stack Exchange Gurus, could a static link be added to auto-import a series of comments into a chat session? I often see output from TeX with the warning overfull hbox, badness 10000. What does this message mean? Can I let my fake flash USB drive get its real capacity through low level format? Recently, I bought an USB flash drive which has a cool design.  The seller said it is 256GB however, I've already known that capacity would be fake. Anyway, I bought it cause of its cool design and want to use this USB flash drive anyway.  What should I do?  I've tried to find a proper tool(vid 1908 / pid 1320) but couldn't. Can I repair this through Low level format? Or partitioning this through Linux? How can I see flags I cast from all sites? Did Dolores Umbridge Have Any Association with Voldemort (or Death Eaters) before His Return? I noticed that Dolores Umbridge was born during the first Wizarding War, so it's very likely she wasn't a Death Eater then (but she is pretty evil -- who knows?).  After that Voldemort was not around in a way that could affect many people, and most wouldn't know he was planning to rise again.  During that time, and up through Voldemort's return (in Goblet of Fire), did Umbridge have any connection with the Death Eaters or with Voldemort?  Was she doing what she did on her own, or was it because of an association with Voldemort or his allies? I need to purchase my first ever lens. So far I've only used lenses that were built into a camera. This specific lens is for the BlackMagic Deign Pocket Cinema Camera (a video only camera, but one that uses photography lenses). I would prefer this to be a prime lens.  I have tried to find the exact way to calculate the focal length of the lens I need but I have not been able to find a good resource to make this simple enough for my limited optics knowledge. Can you please explain a simple way to calculate the focal length of a lens given the following inputs as an example:    I would prefer a simplified mathematical explanation so that I may use it to make similar determinations in the future using a "" method.  I have . Using that method, I came up with:  Required lens focal length = 7.02mm sensor height x (77in / 36in) = 15mm   Is that correct? Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$ Why are all known distributions unimodal? When running a Java app from eclipse my ImageIcon shows up just fine.  But after creating a jar the path to the image obviously gets screwed up.  Is there a way to extract an image from the jar at runtime so I can then open it up?  Or, is there a better way to do this?  I'd like to distribute a single jar file if possible. Capacitor on USB cable? I was messing around with a USB camera. I wanted to change the cable because is trimmed but as I was cutting it I saw a small thing (image below). As I was trying to cut the casing and see what it has inside I saw a spark. I tried to connect it with another cable but I don't know if I'm gonna damage the camera or the USB port.    What is it and can I connect the camera without it? Which is the correct use of these two words, and in which context should one be used rather than the other? I noticed that I got some initial traffic problems at the point where the initial highway is connected to my city. At the beginning I simply elongated both roads that came from that highway, but this doesn't seem to scale.  What I wanted to do now is to create a single six-lane highway from these two four-lane roads. I'd be open to continue to use two roads in parallel here, but I didn't manage to add any way of crossing between the lanes when I tried that idea.  So, how do I merge two one-way roads into a single two-way road? Or, if my idea is really bad, what is a good way to connect the initial highway access to my fledgling city?  Here's a screenshot of the roads I want to connect, if my description wasn't entirely clear: Why are the rooting apps not working anymore? A couple years ago, it was easy to root any device: Install KingRoot, KingoRoot, or a similar app, hit the large ROOT button, and let it do the rest. After 5 minutes or so, a green "Root Succeeded!" prompt will pop up, and we can choose to keep the root app, or uninstall it and install a clean root manager app like SuperSU/Superuser.  How do they work? Why aren't they working anymore? And if they work, should I use them or not? Resources for Blender If I want my app to behave differently on a jailbroken iPhone, how would I go about determining this? GUI/Unity crashing in 16.04 LTS after updates 2018-01-04, compiz segfaults Deciding whether $2^{\sqrt2}$ is irrational/transcendental Is $2^\sqrt{2}$ irrational? Is it transcendental? How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.