Let $A$ be a ring, and let $J$, $K$ be ideals of $A$. The radical of $J$ is the set $$\operatorname{rad}J=\{a \in A:a^n \in J  \text{ for some }  n \in \mathbb{Z}\}.$$ Prove for any ideal $J$, $\operatorname{rad}J$ is an ideal of $A$.   I know that I will start the proof supposing that $a^n \in J$ and $b^m \in J$, and will work to show $(a+b)^{m+n} \in J.$ How would I go about this? and how do I use this to prove the ideal?
$$Rad(I)=\{a \in R | \exists n \in \mathbb{N} \text{ such that } a^n \in I\}$$   R is a commutative ring, I is an ideal.   To show that $Rad(I)$ is an ideal of $I$, we have to show that for $a,b \in Rad(I)$ it stands that $a-b \in Rad(I)$ and for $r\in R$ it stands that $a\cdot r\in Rad(I)$, right??   When $a,b \in Rad(I)$ we have that $a,b \in R$ and $a^n, b^m \in I$.   $a-b \in R$.   But how can we show that $(a-b)^x \in I, x \in \mathbb{N}$ ??
Difference between projection and datum? What's the difference between a projection and a datum? How to draw arrows between circled coefficients of polynomials using tikz After reading Gonzalo's answer to , I thought  I would be able to modify it so that I could have the corresponding coefficients circled and an arrow going from one circle to the next. For example, I would like the coefficient of x^2 on the left, a, circled and the coefficient of x^2 on the right, 5 circled, then a curved arrow drawn between the two circles. How might I achieve this? Also, how would I add some text along the path of the arrow? I reproduce his code here.  \documentclass{article} \usepackage{amsmath} \usepackage{tikz} \usetikzlibrary{calc}  \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}  \begin{document}  \begin{equation}   a\tikzmark{a}x^2 + bx + c = 5\tikzmark{b}x^2 + bx + c.   \begin{tikzpicture}[overlay,remember picture,out=315,in=225,distance=0.4cm]     \draw[-&gt;,red,shorten &gt;=3pt,shorten &lt;=3pt] (a.center) to (b.center);   \end{tikzpicture} \end{equation}  \end{document} Logo is slightly misaligned We built our eCommerce web site using Angular as a single page application.  We have implemented a spinner that runs until the page is responding with the expected content. We have some category pages which were crawled and indexed by Google but which are no longer available on the site. What is the best practice for old pages?  Should I create &quot;404 Not Found&quot; pages for them instead of saying &quot;loading&quot; forever? Is setting a no index meta tag enough? Trigonometric Uncertainty Propagation Gauge fields and strings: Loop equations Force pagebreak after every sentence/paragraph Is there a way to force a pagebreak after every sentence or paragraph?  This may seem like an odd thing to do, but for catching typos and just general editing I find it much easier to look at a sentence or paragraph in isolation so that I am not tempted to skip ahead.  Currently, I just zoom in so I can only see a few word at a time, but this is not a perfect solution. Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$ How to tell Google to ignore URL parameters and forget about any already crawled? arcpy.env.overwriteOutput = True not working for one dataset guess the color of the next card There are 26 red cards and 26 black cards on the table which are randomly shuffled and are facing down onto the table. The host turns up the cards one at a time. You can stop the game any time (even at the beginning of the game). Once you stops the game, the next card is turned up: if it is red, you get $1; otherwise you pay the host one dollar. What is the payoff of this game? How to know if other threads have finished? $f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors Let $f(x,y)=4x^3y^2$  How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most rapidly from the point $(2,1)$? And, what is the rate of change of $f$ in the direction given in the second question?  My Work (What I have done so far):  For the first question: $\nabla f(x,y)=12x^2y^2$i$+8x^3y$j$\nabla f(2,1)=48$i$+64$j$u=(3/5)$i$-(4/5)$j $\therefore D_u f=\nabla f\bullet u=0$  I'm not entirely sure if this is right? Can someone please verify?  For the second question: $u=(3/5)$i$-(4/5)$j $\therefore -||\nabla f(2,1)||=$? How do I find this? And am I doing this right?  I have no idea what to do for the third question. Why should we typedef a struct so often in C? I have seen many programs consisting of structures like the one below  typedef struct  {     int i;     char k; } elem;  elem user;   Why is it needed so often? Any specific reason or applicable area? How Get Parent Field in Attachment? Please Help me...  Account sObject is Parent of Attachment. Now I want to bring Read-only Field Custom field (isReadOnly__c).  SELECT Id, Name, Account.isReadOnly__c FROM Attachment   How It Gets ?  Thanks in Advance  Manu Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer.  In an attempt to make more interesting answers, I ask that this question concern the problem of "Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula for fixed $p\in\mathbb N$".  I've even checked this  without finding what I want.  Rule #1: Any method to compute the sum in question for arbitrary $p$ is good, either recursively or in some manner that is not in itself a closed form solution.  Even algorithms will suffice.  Rule #2: I don't want answers confined to "only some values of $p$".  (A good challenge I have on the side is a generalized geometric proof, as that I have not yet seen)  Exception: If your answer does not generalize to arbitrary $p$, but it still generalizes to an infinite amount of special $p$'s, that is acceptable.  Preferably, the method is to be easily applied, unique, and interesting.  To start us off, I have given my answer below and I hope you all enjoy. I host am ExpressionEngine based website on a single Amazon EC2 instance in the US East (Virginia) region. I use Ubuntu.    The website allows members to post their content, and displays it for other members and for users who are not logged in.   Goal: I need to begin serving the same website across the globe, equally fast to Hong Kong, New York, and Los Angeles.   I considered two solutions:  1.Setting up three separate EC2 instances for three regions: Virginia, Oregon, and Singapore. Use Amazon Rot53 Geo Location to direct traffic to instances based on the browser's IP address.   Pro: the website will load fast for the region.   Contra: these will be essentially three separate websites. Unless I am mistaken?   Setting up one instance in a region equally remote from all three locations: (Ireliand), and boost the server to the highest performance available.    Pro: It will be the same website for all regions.   Contra: it will cost a lot to make it fast enough.   My challenges:   a) I do not know a way to sync server settings, database, and content between multiple Ubuntu instances in real time. If a knowledgeable person could explain to me how to do that, I would be most grateful.   b) I am not thoroughly familiar with all Amazon AWS resources. Does AWS has a built-in method that would allow me to achieve my goal of serving a single PHP / MySQL based membership website across the globe equally fast in all three chosen locations? Strip HTML from Text JavaScript How to replace a video on YouTube JavaScript + Unicode regexes