Given two free modules $R^n$ and $R^m$ over a commutative ring $R$, it's true that $R^n \cong R^m$ if and only if $n=m$. However every proof I've seen of this fact assumes that we've first proven it when $R$ is a field. Although this is probably true (most of us see linear algebra before commutative algebra), this sort of proof is a bit unsatisfactory to me.   Is there a (nice) proof of $R^n \cong R^m$ if and only if $n=m$ that doesn't assume we know it's true when $R$ is a field?  $\bf{Edit}$ Though the linked question implies a proof that does not use the result for $R$ a field, such an implication doesn't mean that the questions are duplicates of one another.
I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold.    This is exercise 2.11 from Atiyah and MacDonald's Introduction to Commutative Algebra, but unfortunately all the solutions available online are either very sparse with regard to this question or seem to use (to my surprise) a generalized version of Cramer's rule. Is there perhaps some other "cleaner" approach to solving this problem?
Minecraft Empty Chests Command Block I am using the following Command to spawn a new chest at the given coordinates with stuff in it at each new round:  /setblock 264 4 997 minecraft:chest 0 replace {Items:[{id:274,Count:1,Slot:0},{id:275,Count:1,Slot:1}]}   The problem with that is that when replacing the old chest all the stuff in the old chest is thrown out. And that's pretty much not what I want. I need to find a way to replace a chest which might have stuff in it without throwing those things out.  Someone else said I should replace the given position for one tick with lava, which I tried. Works in 90% of the tries (in the other 10% the items from the chest are thrown to far away - out of the spawned lava block), but the problem with that is that I can't use it in wooden areas because it might burn everything down.  So what can I do to fix that problem, because otherwise my custom made Pv.P map is not gonna work?  Thanks very much! Create WiFi hotspot on ubuntu Although many questions have already been asked regarding WiFi hotspot, I didn't get any of the solutions working for me.  I connect to the Internet using a 3G Dongle of Huawei 3772 (Vodafone). The connection type is Mobile Broadband connection. I have to share this Internet connection with my other devices like mobile phones and laptops.  How can I create a WiFi hotspot connection for this connection.  I am using Ubuntu 12.04 LTS.  I would like to note that I have already tried some help links from Ubuntu but, they are so confusing that I could not understand them. Spinor dotted and undotted indices Automorphism group of direct product of groups I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups.  Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then $\operatorname{Aut}(G\times H)\cong\operatorname{Aut}(G)\times \operatorname{Aut}(H)$.  I also know that if $G,H$ are abelian and simple and isomorphic, then $\operatorname{Aut}(G\times H)\cong \operatorname{GL}_2(\mathbf{Z}_p)$ where $p$ is the order of $G$.  What I don't know is the following: what if $G,H$ are simple, yet not abelian? Let us for now focus on a more restricted case where $G\cong H$, in particular, $G=H$. What can we say about the group $\operatorname{Aut}(G\times H)$? (I heard something about this group being related to what is called "wreath product", which I do not know...)  Thanks bunch in advance! I was a member of a study team that produced a couple of research papers. The authors of the paper are, for example:      Michael Jordan, Tiger Woods, John Smith and The ABC Study Investigators   In the acknowledgements, it reads:     The ABC Study team consists of members from Harvard University (James Bond, Thomas Harper, David Price, and Jason Jolly)   (Suppose that my name is David Price.)  Can I list such papers in the publication list of my CV? If not, how can I show that I participated in these research papers?  In PubMed, my name was listed as one of 'collaborators.' How to determine exact contrast ratio Because of the recent Web accessibility Law in my region, I need to ensure I'm designing with an appropriate contrast ratio.  WCAG says the value of this ratio is 4.5:1    How do I figure this out in in Photoshop and Illustrator? Did the Big Bang happen at a point? Will Winterbash promote mass spamming due to its short time frame? Is there any concern that , due to their time-limited nature, will encourage mass spamming of posts? Cosmopolitan, for example, can only be achieved by posting to Meta. There are over one million users on Stack Overflow who may want this hat, and have only until January 4th to get it, and their only means of doing so is to invent some reason to post to Meta. If a>0 and b>0,  $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $  What I was trying to do: Suppose a>b. Then, for sufficiently large values of x, $ a^x &gt;&gt; b^x $; so $\sqrt[x]{a^x+b^x} \rightarrow \sqrt[x]{a^x} \rightarrow a $ when $x \rightarrow +\infty$.   Is that idea correct? How can I formalize it? I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation.  I decided to Google this and came upon this article:    It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):  "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."  Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation?  Thank you. How to check file types of uploaded files in PHP? On the PHP website, the only real checking they suggest is using is_uploaded_file() or move_uploaded_file(), . Of course you usually don't want user's uploading any type of file, for a variety of reasons.  Because of this, I have often used some "strict" mime type checking. Of course this is very flawed because often mime types are wrong and users can't upload their file. It is also very easy to fake and/or change. And along with all of that, each browser and OS deals with them differently.  Another method is to check the extension, which of course is even easier to change than mime type.  If you only want images, using something like getimagesize() will work.  What about other types of files? PDFs, Word documents or Excel files? Or even text only files?  Edit: If you don't have  or  and system("file -bi $uploadedfile") gives you the wrong file type, what other options are there? ModernCV Remove horizontal line in sections I want to write my cv with moderncv-Classic style. Can you tell me, how I can remove the horizontal line in the left of each section? Prove recursively defined sequence converges I would like some advice on how to solve problems like the following: Let $(x_n)$ be a sequence defined by $x_1= 3$ and $x_{n+1} = \frac{1}{4-x_n}$. Prove that the sequence converges.  My strategy is to use the Monotone Convergence Theorem, but I am having trouble showing that the sequence is decreasing and bounded below. Here's my work so far:  Decreasing: The first 4 values are $3,1,1/3,3/11$, so let's assume $x_n \leq x_{n-1} \leq \ldots \leq x_1$. Want to show $x_{n+1} \leq x_n$. We have $x_{n+1} = \frac{1}{4-x_n}$. I want to have an upper bound for the RHS, but can't find one and don't really know where to go from here.  Bounded below: I wanted to show that all values are positive, but if we assume $x_n &gt; 0$, that doesn't rule out $x_{n+1}$ from being negative. Merging arrays with the same keys What really allows airplanes to fly? What aerodynamic effects actually contribute to producing the lift on an airplane?  I know there's a common belief that lift comes from the Bernoulli effect, where air moving over the wings is at reduced pressure because it's forced to travel further than air flowing under the wings. But I also know that this is wrong, or at best a minor contribution to the actual lift. The thing is, none of the many sources I've seen that discredit the Bernoulli effect explain what's actually going on, so I'm left wondering. Why do airplanes actually fly? Is this something that can be explained or summarized at a level appropriate for someone who isn't trained in fluid dynamics?  (Links to further reading for more detail would also be much appreciated) How do I create a dynamic key to be added to a JavaScript object variable Let $n\in\mathbb N$ and let $a_1,\ldots,a_n$ be natural numbers smaller than $10^n$. Write each $a_k$ in base $10$ and add $0$'s to the left of each decimal expansion, if needed, so that each $a_k$ is written with $n$ digits. Consider the matrix such that the entries of the $k$th row are the digits of $a_k$ (written in the same order). Let $d\in\mathbb N$ be such that $d$ divides each $a_k$. Prove that $d\mid\det A$.   For instance, suppose that $n=4$ and that your numbers are $3876$, $2784$, $684$, and $8388$, each of which is a multiple of $12$. Then$$A=\begin{bmatrix}3 &amp; 8 &amp; 7 &amp; 6 \\  2 &amp; 7 &amp; 8 &amp; 4 \\ 0 &amp; 6 &amp; 8 &amp; 4 \\  8 &amp; 3 &amp; 8 &amp; 8\end{bmatrix}$$and $\det A=-360$, which is, in fact, a multiple of $12$.  I learned about this problem yesterday. I found it quite cute and I decided to share it with all of you.  Note: I know how to prove it. What is a decoupling capacitor and how do I know if I need one? What is a decoupling capacitor (or smoothing capacitor as referred to in the link below)?  How do I know if I need one and if so, what size and where it needs to go?    mentions many chips needing one between VCC and GND; how do I know if a specific chip is one?   Would an  4-bit parallel access shift register used with an Arduino need one? (To use my current project as an example) Why or why not?  I feel like I'm starting to understand the basics of resistors and some places they're used, what values should be used in said places, etc, and I'd like to understand capacitors at the basic level as well. Resizing videos with ffmpeg/avconv to fit into static sized player I have a html 5 video player which is 700px wide, 400px high. I'm trying to use avconv to use ffmpeg to resize (while retaining the aspect ratio) and making sure it fits into my player.  Input can be a file of any size, so I need to resize the larger ones but center the smaller ones with black bars. So far I've tried: -s and -aspect, I've seen pad being used with the -vf switch but don't understand how it works enough to get what I need.    This is a rough idea of what I need. I'm not sure if it's even possible. It's almost like CSS's max-width/max-height. I realise this may just be a jumble of words but if anyone happens to understand what I'm talking about, I'd appreciate help, Thanks.  avconv command:  avconv -y -i control.avi -vcodec libx264 -b 2000k -bufsize 20M -acodec aac -strict experimental -ar 44100 -ab 256k bigbuck_out.mp4 I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$.  Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x']  \mapsto x'|_U$.  It is clear that this map is onto, as we just take a $u' \in U'$. Then I can use Hahn-Banach to get an appropriate $x'$ such that $x'|_U = u'$ and in particular $||x'||=||u'||$.  Now I need to show that is isometric: since we always have $\|[x']\|\le \|x'\|$ and by Hahn-Banach we have $\|x'\|=\|x'_{|U}\|$, this shows $\|[x′]\|\le \|x'_{|D}\|$ and it should be always true that $\forall y' \in U^\perp: \|x'-y'\| \ge \|x'_{|D}\|$, hence $\|[x']\|\ge \|x'_{|D}\|$. Is this reasoning correct? The isometric property follows.