Let $R$ be commutative ring $R$ with a multiplicative identity $1$. Suppose $p(x)=a_{n}x^{n}+\cdots a_{1}x+a_{0} $ is a polynomial in $R[x]$.   Prove that each $a_0,a_{1},\dots,a_n$ is a nilpotent element in $R$ if and only if $p(x)$ is nilpotent in $R[x]$.   Here is my attempt at the forward direction.   $(\Rightarrow)$ Suppose each $a_0,a_{1},\dots,a_n$ is nilpotent in $R$. Then $a_{i}^{n_{i}}=0$ for each $a_{i}$ and for some positive integer $n_{i}$. Let $N=\prod\limits_{i=0}^{n}n_{i}$. Then $[p(x)]^{N}=0$ which shows that $p(x)$ is nilpotent in $R[x]$.  I think this is correct, but I'm not positive. I couldn't find any concrete examples of a polynomial ring which contains more than one nilpotent element.   For the $(\Leftarrow)$ direction, I was thinking that I should use induction on the degree of $p(x)$. If this is the wrong approach, then I could use some advice on how to prove this.
Let $R$ be a commutative ring.  The nilradical $\text{nil}(R)$ is the set of all nilpotent elements, and it is the intersection of all the prime ideals of $R$.  Is the following true in the polynomial ring? $$\text{nil}(R[X])=(\text{nil}(R))[X]$$  Any polynomial in the right hand side has all its coefficients nilpotent in $R$, hence also in $R[X]$, and therefore it is nilpotent in $R[X]$ as the sum of nilpotent elements.  The other inclusion is not clear to me.  Is there a counterexample?
Based on Logman dictionary:       happen to do something: if you happen to do something, you do it by chance.   for example take a look at these examples:     They happen to be unemployed at the time. I happen to have a degree in computer science.    So based on the definition of the happen to do something, first one means they are probably unemployed at the time. and the second one means I probably have a degree in computer science.     Am I right? Does eating infected flesh have an effect? I don't remember what actually happends when people eat infected flesh. I have in mind :  Gareth and the cannibals eating bitten Bob's leg in . It seems that the cannibals survive for a while after eating Bob's flesh and they seem OK.  The Saviors eating pigs that ate walkers in . We don't know for how long this process has been on, but  has his routine and looks used to doing that. Can we assume that Saviors are eating infected pig flesh for a while now ?  Does it have negative effect in long term ? A disk is stuck in my slot-loading Dell Studio Hybrid. How can I get it out? The slot-loading optical drive in my Dell Studio Hybrid has been acting up. Sometimes it works fine, sometimes it churns and makes some unpleasant sounds. Now I can't even get it to eject the disc that's in there.  Many drives have an "emergency eject" pinhole, but I don't see one here. How can I get the disc out? What are the prerequisites to encounter the legendaries in ORAS? There are a lot of legendaries available in ORAS, however a lot of them will only show up/be available if certain prerequisites are met.  I know Groudon/Kyogre are available through the Story, and      Rayquaza &amp; Deoxys   through the postgame Delta episode, but what about the other available legendaries? What do I need to do to catch them? How to get exact camera view? In MySql's interpreter, it's very easy to dump a table to the screen along with its field names.   There seems to be no simple way to export a table to a tab-delimted or CSV outfile including its column headers.  I'm trying to do this using only SQL or the Linux command line, without writing a program in another language.  Thank you I am trying to define an rgb color. More specifically I want to create this color    which in rgb mode it's (98,190,177).  I tried to define this color using the following code  \documentclass{standalone}  \usepackage{tikz}  \begin{document}  \definecolor{tank}{rgb}{0.098,0.190,0.177}  %\definecolor{tank}{rgb}{98,190,177}  \begin{tikzpicture}   \coordinate (tank) at (0,0);   \draw[red, fill=tank] (-2,0.5)--(-0.5,1)--(0.5,1)--(2,0.5)--(2,-0.5)--(0.5,-1)--(-0.5,-1)--(-2,-0.5)--cycle;  \end{tikzpicture} \end{document}   but the output is totally different from what I am trying to achieve    I also tried to use \definecolor{tank}{rgb}{98,190,177} but I get no color at all. How to properly define an rgb color? Making adult decisions "is" or "are" really not fun: Which is correct? Making adult decisions are/is really not fun.     What is the proper answer and why? My gut says "is" but I can't explain why. Adding a CMS to an existing Magento shop I am working on a project for 3 niche stores built on magento (using magento's multi-store function)  that each get roughly 50k unique visitors a day.  The sites don't currently have a blog or forum or any social networking aspects.    Would like to add a cms to each site that can be centrally run and would like it to take over the front end content from Magento.  Also would like it to maintain an online blog/publication of sorts with videos, articles, and the like with privileges to edit the content given to a dozen or so people with different privileges.  Want to add a forum to each site that is fairly robust and to possibly add some social networking aspects down the road, so extandability and available plugins/mods in each cms is important.  Other than shared login between the forums,blog/publication and store, would like to be able to integrate some content from the forums and blog/publication into the store as well.  After researching this a bit, I am inclined towards Drupal, but I haven't found any modules to integrate it with Magento.  Also, since the blog content will be done by about a dozen nontechnical people, I want something that is very easy to work with.  Lastly, since the site gets a good amount of traffic, speed and security are very important.  What CMS would you recommend integrating in this context?  Deciding between Drupal, Wordpress and ModX.  Also Plone as well.  Thanks. Strange directory under C: of Windows 7 I have been trying to figure out the time complexity of the conjugate gradient method.  I have to solve a system of linear equations given by  $$Ax=b$$  where $A$ is a sparse, symmetric, positive definite matrix.  What would be the time complexity of the conjugate gradient method? Solve the following equation in positive integers $x$ and $y$ Is password confirmation necessary on a Change Password dialog? On one hand there is no need to ask for password confirmation because it is annoying. If I made a mistake when confirming my password I'll need retype my very complex password again. There is also a button that can show/hide the password so I don't need retype it in order to confirm it.   However, some users don't use the password show/hide button and the 'forgotten password' option will be pain for them.  Twitter removed the password confirmation from registration but left the dual confirmation in place for changing password. Is this right approach? What spell slot count formula can I use to base my homebrew mechanic on? At the start of A New Hope are Yoda and Obi-Wan really the last remaining Jedi? I have never read any Star Wars books and have no knowledge outside of the Star Wars films, but after seeing Episode III I am left wondering if there are any other Jedi that escape death and remain in hiding like Yoda did. I mean with regard to the "official" events in the Star Wars universe. How many named characters died in Avengers: Infinity War? How many named characters have died       and/or disintegrated      by the end of Avengers: Infinity War. It is difficult to speculate who died on         Xandar and the Asgardian ship.       So restrict it to on-screen deaths only. Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition Let $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ be a subset of $\mathbb{R}$. I have to prove using the open cover definition that this is compact. Could you help me, please? If $A$ and $B$ are compact subsets of $\mathbb R$, then so is $\frac{A}B$. Given compact $A,B \subseteq \mathbb{R}$ with $\alpha := \inf(B) &gt; 0$, define $$\frac{A}B := \left\{\frac{x}y : x \in A  \text{ and } y \in B\right\}.$$ Show that $\frac{A}{B}$ is compact.  I am thinking of using the following result given in a previous exercice but I don't know how to proceed.     $K \subset \mathbb R^n$ is compact if and only if for all $\{x_{n}\}$ such that $\{x_{n} : n\in \{1,2,\dotsc\}\}\subset K$, there exists a convergent subsequence $\{{x_{\phi(n)}}\}$ that converges to a limit $l \in K$.   When does the infimum come into place? Common binaries with support for command execution, like `awk` and `vim`, not respecting setuid bit? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option.