I am stuck in a question for Algebraic Structures:  I need to prove the following:      Let $R$ be a ring. If $a^2=a$ for any $a\in R$, then $a+a=0$ for any $a\in R$.   (of course $+$, $\cdot$ and $0$ regarding $R$)  P.S. This is similar to a  (if $a \cdot a = 0$ then $a + a = 0$) but I wasn't able to apply the ideas from there. In that question we assumed that $R$ is a unity ring.
I am stuck in a question for Algebraic Structures:  I need to prove: If given any ring $R$, for any $a$ in $R$ $a\cdot a=0$ then $a+a=0$.  (of course $+$, $*$ and $0$ regarding $R$)  I didn't find any question regarding this question.
Aligning three figures next to each other based on text width Problem:  I am trying to align three figures next to each other with text centered to each figure. This is going well until the box for the third box does not adjust after the text width.  Minimal Working Example (MWE):  \documentclass{article} \usepackage{graphicx} \usepackage{caption}  \begin{document}  \begin{figure}[!tbp]   \centering   \begin{minipage}[b]{0.2\textwidth}     \includegraphics[width=\textwidth]{icon-password.eps}     \caption*{Login system}   \end{minipage}   \hfill   \begin{minipage}[b]{0.2\textwidth}     \includegraphics[width=\textwidth]{icon-shopping.eps}     \caption*{Shopping cart}   \end{minipage}   \hfill   \begin{minipage}[b]{0.2\textwidth}     \includegraphics[width=\textwidth]{icon-clock.eps}     \caption*{Temporary information}   \end{minipage} \end{figure}  \end{document}   Outputs:   Desired outcome:  To get the width to adjust after the text so that "Temporary information" can be written out in one line instead of two. In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. How can I re-write the conditionals for this button I've been trying out React for a couple of days and I tried to build a button component with some conditionals. Everything works but I feel like I am not doing it the right way. Could anyone review my code and give me some advice and tell me what I am doing wrong or what i could do better. I think I am using too many return statements. I use Typescript for type checking. Would appreciate any help! import { Link } from &quot;gatsby&quot;;  interface Props {   content: string,   color: string,   path?: string,   type?: string }  export default function Button(props: Props): JSX.Element {   let elementType;    const checkProps = () =&gt; {     const availableColors = ['blue', 'grey', 'dark-grey', 'white'];     const borderColors = [''];     let buttonStyle;      // Check if props.color exists in availableColors     if(!availableColors.includes(props.color)) {       console.error(`&quot;${props.color} color key not available for button style`);       return;     }          buttonStyle = buttonStyles[`button--${props.color}`];      // Check if props.content is valid     if(!props.content) {       console.error(`&quot;${props.content} is not valid as content`);       return;     }      // Check if props.type exists     if(!props.type) {       elementType = &lt;button className={ buttonStyle }&gt;{ props.content }&lt;/button&gt;       return;     }      if(props.type !== 'link') {       console.error(`&quot;${props.type} is not a valid type, use 'link' instead or use nothing for default button`);       return;     } else {         elementType = &lt;Link to={ props.path } className={ buttonStyle }&gt;{ props.content }&lt;/Link&gt;     }   }    checkProps();    return (     &lt;React.Fragment&gt;       { elementType }     &lt;/React.Fragment&gt;   ); }``` So from what I understand transistors are one of the most important electronic elements for computers because of TTL (Transistors-Transistors Logic).  Is this true? If so, how exactly does this work? I have heard the terms "logic gate" and "logic block" tossed around. If  $\mathbb R^*$ and $\mathbb C^*$ be respectively multiplicative groups of non-zero real and complex number then whether $\operatorname{Hom}(\mathbb Z_6,R^*\oplus C^*)$  is  isomorphic to $\mathbb Z_{12}$? or $\mathbb Z_6$? or $\mathbb Z_6\oplus \mathbb Z_2$? How to echo a bang! I tried to create a script by echo'ing the contents into a file, instead of opening it with a editor  echo -e "#!/bin/bash \n /usr/bin/command args"  &gt; .scripts/command   The output:     bash: !/bin/bash: event not found   I've isolated this strange behavior to the .  $ echo ! !    $ echo "!" bash: !: event not found  $ echo \#! #!  $ echo \#!/bin/bash bash: !/bin/bash: event not found    Why is bang causing this?   What are these "events" that bash refers to?     How do I get past this problem and print "#!/bin/bash" to the screen or my file? I made something like this:    I did this with the Ellipse Tool and the Line Tool.  Now the circle is accurate, but the lines inside are very inaccurate. Is there another tool that does the same thing but with a lot more accuracy? Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $. Destructive interference and energy conservation In case of interference, we know, energy is neither destroyed, nor created; but only redistributed. But in the case of an extremely thin film, due to a reflection and hence a phase difference of $\pi$, the film always appears dark due to destructive interference. So, where does the energy go? What are imaginary numbers? At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?  Why is the concept even useful? I have read that positional parameters start at $1 (for example: $1, $2, $3 and so on are positional parameters). But $0 is not a positional parameter.  But why $0 is not a positional parameter?  I think this could be a reason, but not sure:  Positional parameters only take their values when a script is executed. For example: if we do ./myScript Hello, then $1 will have the value Hello. But $0 can take its value on two occasions: when a script is executed (it would have the value of the script name), and when bash itself is executed without a script (it would have the value bash or -bash). I'm just starting to learn linear algebra and I understand Gaussion elimination but I don't understand this proof:    Is the heart of this the fact that all the while when I've been row-reducing matrices to reduced row-echelon form... I've actually been multiplying a matrix by a series of a elementary matrices?  What does the notation $[A\ \ 0]$ mean?  How is the equation $A = E_1^{-1}...E_k^{-1}$ arrived at? Obfuscation is one way, but it can't protect from breaking the piracy protection security of the application. How do I make sure that the application is not tampered with, and how do I make sure that the registration mechanism can't be reverse engineered?  Also it is possible to convert a C# application to native code, and  is too costly.  C# provides lot of features, and is the ideal language for my code, so writing the whole codebase again in C++ is out of the question.  Secure certificates can be easily removed from the signed assemblies in .NET. I was looking at  and I didn't realize that Minecraft had some Easter eggs in it. Besides the Christmas chest Easter egg, and the sheep Easter egg, what else is there? How do I view all available HDD's/partitions? How does AdamW weight_decay works for L2 regularization? hello I have a question about weight regularization in Adam apparently the weight_decay in the AdamW function  has the same impact as an L2 regularization my questions are: is that parameter the same as lambda that we have in the regularization term?  how does it exactly work? and what is its impact on the model complexity? OS X Yosemite brought support for recording iPhone/iPad screen natively without having to using AirPlay applications like AirServer or Reflector.   How do I record the iPhone screen on macOS ? ^x^y unix trick for all instances in last command? One of my favorite Unix tricks is ^x^y, which will take the last command and replace the first instance of "x" with "y". However, I'm wondering if a similar trick works to replace all instances of "x" with "y" in the last command? How to approximate the student-t CDF at a point without the hypergeometric function? Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF in terms of simpler functions?  The t-distribution can   $$t = \frac{X\sqrt{n}}{Y}$$  where $X$ is normally distributed with mean 0 and variance $\sigma^2$; $\frac{Y^2}{\sigma^2}$ has a chi-squared distribution. Maybe this expression can help simplify the t CDF?  I am trying to write an algorithm to compute the student-t CDF for any real number, so using the hypergeometric function will be extremely difficult. I am also trying to avoid integrating the PDF to get the CDF. Can a wizard under the effects of Feign Death issue telepathic commands to their familiar? A wizard has a summoned familiar within 100 feet that is currently idle and has  (PHB, p. 240) cast on them by another PC. Would the wizard still be able to issue telepathic commands to their familiar?  Specifically, while Feign Death states that they appear dead and are incapacitated and blind, they aren't listed as being unconscious.   doesn't state that issuing telepathic commands requires an action (which you can't do while incapacitated), whereas it does state that seeing through your familiar's eyes requires an action as does dismissing it.  I was DMing a game where this occurred and I hastily ruled that the wizard was unable to issue commands, but I am second guessing myself after the fact. And knowing my PCs this situation is likely to happen again.