This is in connection with the question: . I have found a solution without using commutativeness. I don't understand why commutativeness will be necessary to prove it. Here is my solution:  If possible let char $R = pq$ where $p,q\in\mathbb N$ such that $p,q&gt;1.$ There exists $a,b\in R$ such that $pa\ne0,qb\ne0.$ Then $(pa)(qb)\ne0,$ a contradiction since $(pa)(qb)=(pq)(ab)=0.$ Thus char $R = 0$ or prime.  Please tell me where did I go wrong.
Question: Suppose that $R$ is a commutative ring without zero-divisors.   Show that the characteristic of $R$ is either $0$ or prime.   I have established that every element in a commutative ring $R$ without zero divisors have the same additive order $n$.   Now, if no such additive order n exists, then the characteristic of $R$ is $0$.  Obviously, if a finite additive order exists, Char of $R$ is finite.  How do I show that Char of $R$ is prime? It probably involves lagrange's theorem and the order of the element in $R$.  Hint is appreciated.  Thanks in advance.
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ How do I calculate a time delta in my Google Spreadsheets? Is there a ZIP code I can enter when paying-at-the-pump in the USA with a foreign credit card? The American convention in quotations is (typically) to place punctuation inside quoted text. But I always run into situations where the punctuation of the quote interferes with the punctuation of the sentence. How would you punctuate this (American, non technical)?     When my friends ask, "What do you want for your birthday?", I never know how to respond.   It seems odd to place the last comma outside the quote simply because of the question mark. Is that the preferred (i.e. most often accepted) standard? How do I test a private function or a class that has private methods, fields or inner classes? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. "He saw it stop and his son get out" I have been reading Men With Brooms for a few days. I read a sentence which did not make any sense to me, so I landed here to get some help on it. I have pasted the sentence from the novel. Please go through it and let me know your valuable feedback on it.  An excerpt from the novel (Men With Brooms: A Sweeping Epic, paperback 2002, a novelization by Diane Baker Mason):     Gordon was about to walk away from the Impala when he saw it stop and his son get out. So it was real the boy had come.   As per my opinion it should have been:     Gordon was about to walk away from the Impala when he saw it stopped and his son got out. So it was real the boy had come.   I have two questions here:   Why did the writer use stop and get instead of stopped and got? As she is telling us a story which happened in the past. Let's say the writer is telling us her mind's situation and she used the present tense for it but why did she use stop and get instead of stops and gets? Using PTSerif-TLF for Cyrillic with TeX Gyre Pagella I originally read this book in the mid- to late '70s.   The premise was a bureaucracy whose job it was to manipulate events in human history by making the smallest possible change that would effect the desired outcome. Not time cops in the sense of going after law breakers.  The intent was to maintain a policy of non-interference as much as possible but when interference was warranted then to make the least possible change and to do so invisibly.  For example, they might take someone's keys off the dresser and put them just out of sight underneath the dresser where they plausibly might have fallen.  The subject would never know of the manipulation but would then be delayed getting out of the house and the ripples grew from there to avert a war or otherwise affect events on a large scale.  After reading that, whenever I would lose something - especially in an improbable way such as dropping a screw at my feet and then finding it on the other side of the room half an hour later - I would take some consolation in the thought that I had just averted some global catastrophe.  I was pretty scatterbrained as a kid (we call that ADHD nowadays) so I experienced this a lot.  Given that I've spent so much time pondering the ideas from the book, I'd love to re-read it. We have a pool of N number of balls.  There are: A number of white balls (W), B number of green balls (G), C number of blue balls (B), and D number of pink balls (P)  In how many ways (combinations) we can pick X number of balls from this pool? Is there a formula for something like this?  This is similar to combinations with replacement, but there are limitations to how many times a certain object can be replaced. Landscape printing from HTML I have a HTML report, which needs to be printed landscape because of the many columns. It there a way to do this, without the user having to change the document settings?  And what are the options amongst browsers. How to create a link to a URL provided in a bib file with custom text Using biblatex, how can I get a URL from the bib file so that I can use it in the \href command?  I am guessing that the \usefield command is what I am looking for and that in order to use that I would need the \entrydata command. But both are undefined.  \documentclass{article} \usepackage[backend=biber]{biblatex} \usepackage{hyperref}  \usepackage{filecontents} \begin{filecontents*}{sources.bib} @online{biblatex,     url = {http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf},     urldate = {2018-02-02}, } \end{filecontents*}  \addbibresource{sources.bib}  \begin{document}      % This would work, but I don't want to duplicate the URL from the bib file:     \href{http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf}{biblatex documentation}      % Does not work because \citeurl{biblatex} is already a link, not the URL:     %The \href{\citeurl{biblatex}}{biblatex documentation}      % Does not work because \usefield and \entrydata are undefined:     %The \entrydata{biblatex}{\usefield\href{url}{biblatex documentation}}      \printbibliography \end{document} What conditions are necessary for a Wither Skeleton to spawn naturally, and can I manipulate those conditions to increase the chance of them spawning?  I want to make a Wither Skeleton spawning pad in the Nether but I'm having trouble finding Wither Skeletons. I've only heard that they spawn in Nether Fortresses. I already read it from manual but I can't see difference..  su - change user ID or become superuser  sudo -s [command]   The -s (shell) option runs the shell specified by the SHELL environment variable if it is set or the shell as specified in passwd(5).  If a                    command is specified, it is passed to the shell for execution.  Otherwise, an interactive shell is executed.  sudo -i disappear description in manual Add parenthesis as an accepted Markdown ordered list delimiter The Markdown editor knows that an empty line, followed by a number, followed by a period, followed by a space, is a list item.  This works great It's almost magic  The problem is that I almost always use a close parenthesis for my lists. I think it's a pretty common practice, but the problem is that the editor / Markdown parser doesn't accept parenthesis as an acceptable list delimiter:  This fails horribly This is on the next line I swear Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$ $(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there. polyglossia or bidi bug: [\d\.] gets inverted to [\.\d] After bidi was updated to v. 19.6 (2016-05-23), using at least one RTL language with polyglossia leads to number and following dot being inverted. Cf.  \documentclass{book} \usepackage{polyglossia} \setotherlanguage{hebrew} \begin{document} 6. \end{document}   with and without \setotherlanguage{hebrew}. Note that it is not necessary to actually use that language.  I do not know whether this is a bug in bidi or an incompatible change which polyglossia needs to adapt to. Because of this I have not opened an issue on polyglossia's github page. Sadly, bidi's contact information are outdated – the sourceforge referred to on CTAN is unused and the github page that is mentioned in the package's readme and PDF does not seem to exist any more either.  The problem might be something along the lines of     Does anybody have the previous version of bidi at hand?  TIA, Dario When I add the picture, the word and pdf is the full page and the Latex moves to the right of the page. How can I solve this problem?  this is my work. this is my work and pdf link: https://we.tl/t-cgDSwDuu8H  I couldn't stop the pictures from sliding right. What should I do to make the image look like the original PDF file? If I reduce the scale of the picture is centered but this time it becomes very small.  Is a situation caused by page margins?  \documentclass [12pt, a4paper]{article} \usepackage{color} %\usepackage[turkish]{babel} \usepackage{graphicx} \usepackage{tabularx} \usepackage{multirow} \usepackage{graphicx} %\usepackage[pass,showframe]{geometry} %\usepackage[T1]{fontenc}  \begin{document}  \definecolor{renkbir}{RGB}{0,32,96}  \begin{tabular}{ m{8,5cm} m{6cm} } \Large{\textbf{MsdasdasdN }} \large{\textbf{\textcolor{renkbir}{Firma Adı }}} &amp;\small {Uçak Mühendisi \newline T.C No: asd \newline asd \newline asd \newline asd} \end{tabular} \ %Uçak mühendisi Tc No kısımını sağa yaslamak istedim fakat yapamadım.  \begin{center} \large{\textbf{\textcolor{renkbir}{KİŞİSEL BİLGİLER}}} \end{center}  \begin{center} \begin{tabular}{ l l r } Ad &amp; Mehm &amp; \multirow{9}{*}{\includegraphics[scale=0.15]{2.pdf}} \ Soyad &amp; asdasd \ Doğum Tarihi &amp; sanane (23 Yaşında) \ Adres &amp; asd \ Telefon &amp; asd \ Mail &amp; asd \ Sigara Kullanımı &amp; Hayır \ Ehliyet &amp; B \ Askerlik &amp; Tecilli (31.12.2020) \ \end{tabular} \end{center}  %\vspace{1cm} Burası boşluk bırakma komutu.  \begin{center} \begin{tabular}{ m{4cm} m{6cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{EĞİTİMLER}}}} \ \hline \multirow{2}{}{Üniversite(YL)} &amp; asd \newline Makine Mühendisliği \newline Özel Öğrenci / Türkçe \newline Durumu : Devam ediyor &amp; \multirow{2}{}{Eylül 2018 / } \ \hline \multirow{2}{}{Üniversite} &amp; asd \newline Uçak ve Uzay Bilimleri Fakültesi \newline Uçak asd \newline Örgün Öğretim / İngilizce \newline Durumu : Mezun &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \hline \multirow{1}{}{Lise} &amp; asd \newline Anadolu Lisesi (100 / 90,02) &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{3cm} m{4cm} m{4cm} m{4cm} } \multicolumn{4}{c}{\large{\textbf{\textcolor{renkbir}{YABANCI DİLLER}}}} \ \hline \textbf{İngilizce} &amp;\textbf{Okuma:} İyi &amp;\textbf{Yazma:} İyi &amp;\textbf{Konuşma:} Orta \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{5cm} m{5cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{BİLGİSAYAR BİLGİLERİ}}}} \ \hline \multicolumn{3}{l}{Donanım - Ofis Programları} % donann ofis koyu yazmak istedim yapamadım. \ Ms Excel &amp; Çok İyi &amp; +5 Yıl \ MS Word &amp; Çok İyi &amp; +5 Yıl \ Ms Powerpoint &amp; Çok İyi &amp; +5 Yıl \ \hline \multicolumn{3}{l}{Yazılım Geliştirme} % donann ofis koyu yazmak istedim yapamadım. \ Office &amp; Orta &amp; 2-3 Yıl \ Matlab &amp; Orta &amp; 2-3 Yıl \ Phyton &amp; Orta &amp; 0-1 Yıl \ Fortran &amp; Başlangıç &amp; 0-1 Yıl \ C++ &amp; Başlangıç &amp; 0-1 Yıl \ Ansys &amp; Başlangıç &amp; 0-1 Yıl \ Miktex &amp; Başlangıç &amp; 0-1 Yıl \ \hline \multicolumn{3}{l}{Bilgisayar Destekli Tasarım} % donann ofis koyu yazmak istedim yapamadım. \ SolidWorks &amp; Orta &amp; 2-3 Yıl \ AutoCAD &amp; Orta &amp; 0-1 Yıl \ \end{tabular} \end{center}  \begin{figure} \includegraphics[width=\linewidth]{2.pdf} \end{figure}  \begin{figure} \includegraphics[scale=1]{2.pdf} \end{figure}  \end{document}   Latex SS is here:  PDF SS is here: Correlation with a constant Related:  (not a dupe, that's a feature request and was closed as a dupe of another one)        (old) post does indeed say that Link-only answers can be flagged as NAA, but there are a lot of posts like  which define Not An Answer is a way that it doesn't apply to link-only answers.  Can we get some up-to-date consensus on this? That link only answers are not allowed is . What is the value of $\arctan(1/2)+\arctan(1/5)+\arctan(1/8)$?