How do I approach this question?   We know that $A$ is an ideal of $R$ and $r-s \in A$ where $r, s \in A$. Now, where do I go from here?
How to go with proving this : Let $R$ be a ring and let $I$ be an Ideal of $R$.  Prove that the factor ring $R/I$ is commutative iff $rs-sr \in I$ $\forall r$ and $s$ in $R$  any hint how to go with this...
Issue in evaluating the performance of my "clustering algorithm" using NMI, ARI when the "ground truth" is available? I've got a rather large number of files on my Windows drive that have a colon in their name. (These came from unpacking some Unix archives.)   When I try to delete them, Windows (XP) complains that the file does not exist, and refuses to delete it. This happens when trying to delete or rename it from explorer or the command line. Still, chkdsk does not complain about those files nor fix the problem.   Any ideas on how to get rid of them? Given that , I wonder if every intelligent biological species is capable of producing beings that are Force-Sensitive?   Or are there some species characteristics that can make it 100% impossible for them ever to produce a Force-sensitive individual.  NOTE: Let's exclude - if they exist in EU - things like random new-species-creating mutations - e.g. the individual should be 100% the same species, able to interbreed. The link here:  shows that $\max(a,b)=\frac{a+b+|a-b|}{2}$.      Is it true that $\max(|a|,|b|)=\frac{1}{2}(|a|+|b|)$? Is there a way that I can configure Visual Studio 2008 to understand CamelCase?  Specifically, I'd like to be able to get Ctrl + right or left cursor to take me to a subsection of a variable or type name. i.e., if my cursor was at the start of this line: LongNamedExampleClass longNamed = new LongNamedExampleClass();  and I hit Ctrl + right, I'd like to jump to &quot;Named&quot; in the class name, rather than the start of the variable name.  If I hit it again, I'd like to go to &quot;Example&quot;.  I hope that makes sense. I used this navigational feature in Eclipse quite a lot, and find it difficult to unlearn something so ingrained. PCA and proportion of variance explained Why do non-hydrogen atomic orbitals have the same degeneracy structure as hydrogen orbitals? running script with ". " and with "source " I was wondering if the following two ways of running a bash script are equivalent?  . ./myScript.sh  source myScript.sh  Are they both running the content of the script instead of running the script, i.e. not creating a subshell for running the script? Install on usb external hard drive I am new to ubuntu and i would like some help. I have a laptop with windows 7 installed but i would love to have ubuntu too because it's a great OS. So i would like to install ubuntu on an external usb hard drive (320 - 500 GB) and i have some questions about that :   Do i need a usb external hard drive which will be compatible with ubuntu - linux ? or i can do it with any hard drive  If i make a partition on it for ubuntu staff will i be able to use it and save other staff in the usb hard drive (movies,songs,documents etc) ? If i have ubuntu installed on the usb hard drive will i be able to boot ubuntu from any pc ? Are my changes going to be saved normally or my settings will be lost after shut down ?   Waiting for your answers, if something is not explain well tell me to try explain it better . Thank you! If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true. public static map&lt;id,RecordType&gt; MapRecordtype = new Map&lt;id,RecordType&gt;([SELECT id, DeveloperName FROM RecordType where sobjecttype ='move__c' and                                         Developername in('Assignment','Draft','Domestic_Draft','LTST_Draft','MX_Domestic','MX_Domestic_draft','Termination','International_Relo','Localization','J1',                                        'Back_To_Back','Repatriation','PAM','US_Domestic_Relo','CHN_Domestic_Relo','CHN_Domestic_Draft')])   Can we write schema for above query? Prove that $6$ divides $n(n + 1)(n + 2)$ I am stuck on this problem, and was wondering if anyone could help me out with this. The question is as follows:  Let $n$ be an integer such that $n ≥ 1$. Prove that $6$ divides $n(n + 1)(n + 2)$. Note: An integer $a$ divides an integer $b$, written $a|b$, if there exists $q ∈ Z$ such that $b = qa$. Alternatively, $a|b$ if dividing $b$ by $a$, $b ÷ a$, results in an integer.  Should I do a proof by induction? All help/input is appreciated! "Home page" or "homepage"? Is there a convention for the spelling of the name of the main page of a website? Should it be home page, with a space between the two words; or homepage, all one word? Hat duplicates when info is expanded It seems like there is a bug that is causing a duplicate hat to appear when I expand info, like pressing &quot;(more)&quot; on a suggested edit. This bug does not seem to be specific to the suggested edits review queue though, as others have reported seeing this bug in other places, like code snippet expansion. Steps to repro:  Put on a hat. Navigate to one of your own suggested edits. (You can just also open the Suggested Edits review queue and go to a random suggestion, but there is no guarantee any of those users will be wearing a hat.) Click the &quot;(more)&quot; button to expand the top box.  What should happen: Hat should shift down with the rest of the content. What happens: All the content shifts down, but a duplicate hat remains in the initial place. Before expansion:  After expansion with duplicate hat: What is the difference between    a parameter passed by reference a parameter passed by value?   Could you give me some examples, please? Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$  I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.  Now I have the next estimation:  \begin{align*} \|f_a-f\|_1 &amp;= \int_{\mathbb{R}} |f(x)-f(x-a)| dx \\ &amp;= \int |f(x)-g(x)+g(x)-g(x-a)+g(x-a)-f(x-a)| \\ &amp;\leq \|f-g\|_1 + \|f_a-g_a\|_1 + \int |g(x)-g(x-a)| \end{align*} my question is: I can argue that $\lim_{a\to0} \int |g(x)-g(x-a)| = \lim_{a\to0} \lim_{T\to\infty} |g(x_0)-g(x_0-a)| 2T$, now I think that I can change the order of the limits here, but I am not sure why?  P.s $x_0$ is some point in $[-T,T]$, and the above is valid from the intermediate integral theorem for continuous functions, right?  Thanks. Typing strange letters¿ w/o numpad? How do i type ♣ (alt+5) and ♥ (alt+259) and other characters like it on a keyboard w/o a numberpad? using the numbers 1-9-0 doesnt activate those characters. Note i want to TYPE and not use that windows program to select/copy/paste One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful concept, it hardly explains the long stability of the solar system. Normally an N body problem with inverse square mutual interaction is an example of a chaotic system. Is there any real progress about this stability issue? How to use `ybar interval` without skipping the last value Consider the factorization into irreducibles of $6$ in $\mathbb{Z}[\sqrt{-5}]$. We have $6=2 \times 3$ and  $6=(1+\sqrt{-5}) \times (1-\sqrt{-5})$, i.e. $2$ distinct factorizations. And,  $$6^2=3 \times 3\times2\times2$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5}) \times (1+\sqrt{-5}) \times (1-\sqrt{-5})$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5})\times3\times2.$$ More generally, $6^n$ will have $n+1$ distinct factorizations into irreducibles in $\mathbb{Z}[\sqrt{-5}]$ by a simple combinatorial argument. But, can we construct a ring in which there exists an element that has an infinite number of distinct factorizations into irreducibles? To make life harder, can we construct an extension of $\mathbb{Z}$ in which this happens? I have been thinking about this for a while and have managed to find no foothold.. Any help is appreciated.