Viewing $m/m^2$ as a vector space I am reading Atiyah MacDonald`s Commutative Algebra textbook. In Page 91, next paragraph, I do not understand part of the following:      If $A$ is a local ring, $m$ its maximal ideal, $k=A/m$ its residue field, the $A-$module $m/m^2$ is annihilated by $m$ and therefore has the structure of a $k-$vector space.    I understand that $m/m^2$ is an $A-$module. I do not get how $m/m^2$ is annihilated by $m$. Why does one need this statement (i.e. annihilation by $m$) to conclude that  $m/m^2$ is a $k-$vector space ? I mean $k$ is a field, so  $m/m^2$ is automatically a vector space as it is a $k-$ module. Finally why is $m/m^2$ Special and not $m/m^n$ for $n&gt;2$?
Why an R-module is an R/I-module precisely when it is annihilated by I? Let $I$ be an ideal of $R$ and let $M$ be an $R$-module. Prove that the formula $(r + I) \cdot m = r \cdot m$ for $r + I \in R/I$, and $m \in M$ makes $M$ into an $R/I$-module if and only if $i \cdot m = 0$ for all $i \in I, m \in M$.   I don't quite understand why we must have $i \cdot m = 0$ instead of $I \cdot m \in M$. This is for proving the only if part.
Integrating MySQL with Python in Windows I want to have a page-filling and -breaking longtable, so I decided to try xltabular. Unfortunately xltabular seems to screw up the booktabs rules in the normal tabular environments, see:  \documentclass{article} \usepackage{booktabs, xltabular} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tabular} \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}     Not loading xltabular resolves the issue.  Did I miss some package option or some incompatibilities? Is there a better package than xltabular which doesn't break booktabs?  (I am using TeXLive 2019 and compile with lualatex)    EDIT (by Skillmon)  I reduced the bug to the following MWE (which doesn't make much sense on its own), it looks like booktabs tries to be smart detecting something, but fails badly.  \documentclass{article} \usepackage{longtable} \let\xltabular\relax \usepackage{booktabs}  \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}   It seems to suffice that longtable is loaded and \xltabular is defined (even if it's just \relax). Differential equation for Harmonic Motion Particle undergoes simple harmonic motion.  Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$   Compute displacement and acceleration when the velocity is square root of $8$.  I am not familiar with this kind of application question.  So, its displacement is $1$, does that mean that $y=1$ when $x=0$ ? Dynamically set value of a file input Quick question to anyone who has used this LaTeX template:    Any ideas on how to change the left margin to get more words onto one page? Spice containers purchase  I'm planning on upgrading my spice storage. Reading some advice online (e.g. , and ) I came to the conclusion that spice containers should:   Not be exposed to direct sunlight and high temperatures Be airtight   Which means that either you store it inside of some closed cupboard/wardrobe/room or you have containers which do block light and are in a relatively shady place. And the containers need to be airtight.  Available solutions  I've been browsing for some containers available online in my country and the 2 most sensible options I've seen are these:       The problem  The first one is hermetic (it has some rubber around the lid) but it's clear glass. The other one does not have the rubber, just a regular twist-off lid, but it's amber glass.  It'll be hard for me to find any space inside of cupboards to store all of my spices (I need at least 30), so I'd rather buy amber glass container, which I believe will effectively block the sunlight, even when kept on the counter. But I can't find any hermetic amber-glass containers, only ones with regular lid.  The question  Is that regular twist-off lid enough to keep spices airtight and fresh? Or should I focus on buying a hermetic container (like the one in clear glass). I could then for example paint them black to block the sunlight. Calculating $\lim_{x\to+\infty}(\sqrt{x^2-3x}-x)$ Let $f(x) = \sqrt{x^2-3x}$ and $g(x) = x$. Calculate the following limits, showing all working.  I've done the first two -   $$\lim_{x\to0}f(x)\\ =\lim_{x\to0}\sqrt{x^2-3x}=0$$  $$\lim_{x\to-\infty}\frac{f(x)}{g(x)}\\ =\lim_{x\to-\infty}\frac{\sqrt{x^2-3x}}{x}=\sqrt{1-\frac{3}{x}}=1$$  How do I calculate this one?  $$\lim_{x\to+\infty}(f(x)-g(x))$$ Can the 'bin size' in a histogram be thought of as a regularity constraint? How can I edit my network profile independently? My oldest account on the network is Area 51, followed by Mi Yodeya.  I recently filled out the "about me" section on Mi Yodeya (but nowhere else) and did not choose the "update all sites" option because my blurb is pretty specific to that site.  I intended to have that blurb on exactly one site.  I was surprised tonight to discover that that blurb is now on my network profile, where I see no "edit" link.  Am I stuck with either having a site-specific blurb as my network blurb (because it's my first non-Area-51 site) or rewriting it to be more general?  Or is there some way that I can independently edit the network profile?  (Or designate a different site as the default?)   shows that someone else had this problem too, but I'm asking for support, not for a new feature, so I'm asking this separately. How to format table with long column head entries? for (i=0 ; i&lt;=10; i++) {     ..     .. }  i=0; while(i&lt;=10) {     ..     ..     i++; }   In for and while loop, which one is better, performance wise? Does the use of words such as 'fail', 'wrong', 'incorrect' and 'invalid' have a sufficiently negative impact on a users frame of mind to warrant the prohibition of their inclusion in the error messages generated by typical web forms?  i.e. Should we rephrase examples like these to just use positive words?  'date entered incorrectly'  or  'failed to complete registration, please see below for invalid field entries'  (PS, the examples are there to show use of the words in an error message, not to serve as examples of error messages I would use or things that need correcting in answers, I'm not looking for advice on how to write form error messages, but for information about the effect of certain words on a users frame of mind) Were I rich, I would live on Long Island.   If I were rich, I would live on Long Island.   Is the first sentence still used, or is used in particular contexts (in example, to give emphasis to the sentence)? Elantech Touchpad not working on Ubuntu 16.04 and Arch Linux So I have and Asus R558UF notebook with an Elantech Touchpad. After installing Ubuntu 16.04, the touchpad worked once but after restarting, it failed to work and has not been working ever since.  This is output of xinput:   ↳ Elan Touchpad                            id=12   [slave  pointer  (2)]   In Mouse and Touchpad settings, the touchpad is enabled. What should I do to fix this? Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$ I came across the following problem in my self-study of real analysis:     For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = \frac{1}{2}(a+b-|a-b|)$$   So $a \geq b$ iff $a-b \ge0$ and $b \ge a$ iff $b-a \ge 0$. At first glance, it seems like an average of distances. For the first case, go to the point $a+b$, add $|a-b|$ and divide by $2$. Similarly with the second case.   Would you just break it up in cases and verify the formulas? Or do you actually need to come up with the formulas? How can I create an executable JAR with dependencies using Maven? What updates should be done to older monsters to bring them in line with MM3 era monsters? I know that to better balance paragon and epic level players against similar level monsters, WotC changed the math used for monsters for MM3, and I believe that they changed some of the types of powers they gave monsters.   If I want to update monsters from MM1 or MM2 to be in line with MM3 era monsters, what changes should I make? 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. How to use iSight webcam without activating the green light? Recently a the Dutch government advised to put a tape on your webcam to prevent hackers from taking pictures and video using your device ().  The build in iSight camera in Apple laptops have a green light next to the camera, this light turns green when you use the camera.  Is it possible to use the camera without the activation of the green light? Personally I think the green light turns on when the sensor requires power and this is hardwired in the hardware. Am I wrong?  I have found similar questions on stackexchange, but I found non that answer this specific question. Get all non-unique values (i.e.: duplicate/more than one occurrence) in an array