Let $A$ be a integral domain, $b\in A$ a nonzero element and $S=\{b^n\mid n \in \mathbb{N}\cup\{0\}\}$ Prove: $S^{-1}A \cong A[x]/(bX-1)$ For this we define the homomorphism $\varphi: X\longrightarrow1/b$ and prove that $\ker\varphi=(bX-1)$ Since $\varphi(bX-1)=b·1/b-1=0$ we have that $\ker\varphi \supseteq(bX-1)$ Now we have to see that any $f\in \ker\varphi$ is in the ideal $(bX-1)$, the obvious way to approach this is to divide $f$ by $bX-1$, but we can't apply the division algorithm directly since the main coefficient of $bX-1$ is not necessarily $1$. Any help to prove $\ker\varphi \subseteq(bX-1)$?
Let $R$ be a commutative ring and $A=\{1,a,a^2,\dots\}$ for some $a\in R$. Prove that $A^{-1}R$ is isomorphic to $R[T]/(aT-1)$.    I guess I'm meant to find a surjective homomorphism between $A^{-1}R$ and $R[T]$ and then use first isomorphism theorem. What homomorphism should I use?
I'm trying to convert some strings that are in French Canadian and basically, I'd like to be able to take out the French accent marks in the letters while keeping the letter. (E.g. convert é to e, so crème brûlée would become creme brulee)  What is the best method for achieving this? Does $\{f_ng_n\}\to fg$ uniformly? Set root domain record to be a CNAME I need to create an NS record for a domain that is a CNAME, for the purpose of having two domains pointed at one IP, and not having to maintain the current IP address in two different places.  The DNS provider for this domain is DynDNS, but they block this operation:     CNAME cannot be created with label   that is equal to zone name   I can do this with another domain whose DNS is served by 1and1:  root@srv-ubuntu:~# dig myseconddomain.co.uk  ; &lt;&lt;&gt;&gt; DiG 9.4.2-P1 &lt;&lt;&gt;&gt; myseconddomain.co.uk ;; global options:  printcmd ;; Got answer: ;; -&gt;&gt;HEADER&lt;&lt;- opcode: QUERY, status: NOERROR, id: 61795 ;; flags: qr rd ra; QUERY: 1, ANSWER: 2, AUTHORITY: 0, ADDITIONAL: 0  ;; QUESTION SECTION: ;myseconddomain.co.uk.             IN      A  ;; ANSWER SECTION: myseconddomain.co.uk.      71605   IN      CNAME   myfirstdomain.co.uk. myfirstdomain.co.uk.     59      IN      A       www.xxx.yyy.zzz  ;; Query time: 298 msec ;; SERVER: 10.0.0.10#53(10.0.0.10) ;; WHEN: Tue Aug 18 14:17:26 2009 ;; MSG SIZE  rcvd: 78   Is this a breach of the RFCs or does DynDNS have a legitimate reason for blocking this action?  Followup Thanks to the two answers already posted I now know that 1and1 IS breaching RFCs to do this. However it does work and they seem to support it. For a company that hosts so many domains it seems very odd that they get away with doing this on such a massive scale without objection.  More followup  The output of "dig myseconddomain.co.uk ns" as requested.  root@srv-ubuntu:~# dig myseconddomain.co.uk ns  ; &lt;&lt;&gt;&gt; DiG 9.4.2-P1 &lt;&lt;&gt;&gt; myseconddomain.co.uk ns ;; global options:  printcmd ;; Got answer: ;; -&gt;&gt;HEADER&lt;&lt;- opcode: QUERY, status: NOERROR, id: 18085 ;; flags: qr rd ra; QUERY: 1, ANSWER: 2, AUTHORITY: 0, ADDITIONAL: 2  ;; QUESTION SECTION: ; myseconddomain.co.uk.             IN      NS  ;; ANSWER SECTION: myseconddomain.co.uk.      4798    IN      NS      ns67.1and1.co.uk. myseconddomain.co.uk.      4798    IN      NS      ns68.1and1.co.uk.  ;; ADDITIONAL SECTION: ns67.1and1.co.uk.       78798   IN      A       195.20.224.201 ns68.1and1.co.uk.       86400   IN      A       212.227.123.89  ;; Query time: 59 msec ;; SERVER: 10.0.0.10#53(10.0.0.10) ;; WHEN: Wed Aug 19 12:54:58 2009 ;; MSG SIZE  rcvd: 111 Could authors overturn recent community review decisions? When new users edit a post, the review is posted to a review queue and other reputable users can review the edit and choose to approve or reject it. However, I feel that a lot of people do it pretty quickly.    to . He was fixing broken links in the answer, and as soon as I saw it I knew it would have been a good edit. However, only one reviewer found so, and three others rejected it. I ended up manually doing the edit, and poor Onato didn't get any credit for it (in fact, he got an undeserved 'rejected' for it).  The point I want to bring is that, first and foremost, the person who wrote the answer probably knows best if an edit is acceptable or not. Could there be a time frame where the author of the answer (or question) can overturn a review decision?    EDIT: one more  was rejected by overeager reviewers as well. I absolutely disagree with the notion that the community knows better. The community repeatedly proves that it only wants Internet points.    EDIT: I told a commenter to fix something himself in one of my posts, but . I went back and applied the changes myself. It's not credited to him and his stats got a rejected edit now (which really should be an approved edit). The reason ("This edit is incorrect or an attempt to reply to or comment on the existing post") is blatantly wrong and shows once more that the community has much less insight into edit quality than the people who wrote the answers in the first place. It's true that the UI didn't make it easy for them, but those guys clearly didn't read the comment thread; I did, because as the author, I was notified.    EDIT: This time around, a user fixed a blatant (though small) mistake in one of my accepted answers. One hour later, another, unrelated 10k user , even though it had been approved by the community. I went back to undo his undo.  This goes on again to show that an active user is the best judge for the quality of an edit, and that since I can have the last word on edits on my own posts, I should be allowed to retroactively overturn a community edit vote and accordingly credit the person who proposed the change.    EDIT I wrote a  years ago. Some guy found it and tried to fix it but the community thought that it wasn't worth it. I think that the changes are great and once more I'll integrate them by myself into my post. Unfortunately for this user, he won't be credited for it. I still don't understand why I can't have a binding vote on edits on my posts. Isomorphism involving Eilenberg-Maclane space, Tors. Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); A)$ and $\text{Tor}_*^{\mathbb{Z}[\pi]}(A, \mathbb{Z})$? Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded. Give an example of a continuous function $f : [0,  ∞) \mapsto [0, ∞)$ such that  $\int_{0}^{\infty}f(x)dx$ exists but  $f$ is unbounded.   I have been thinking about this. And I have come to the conclusion that I will need to construct a function, $f$, such that $f$ is a sequence of triangles of increasing height, but decreasing base. I obviously need $f$ such that both the height of the triangles, and sum of the bases tends to infinity. But I also need that the $\sum (\text{height} \times \text{base}) \leq \infty $ Who is this running kid wearing a shirt with Uchiha symbol? There's a kid wearing a shirt with Uchiha symbol, but I have no idea who s/he is.    Who is this kid? Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ I've installed phpmyadmin sudo apt-get install phpmyadmin. Of course, I have also mysql and a web server running properly.  Now how can I reach it? Windows 8.1 fresh install with Windows 8 licence Two players shooting a target alternately Which meta am I supposed to post on? I'm mostly active on  and we have a separate . Most of my questions on IT Security meta have answers with links here. So my question is, where am I supposed to post what i.e. feature request, bug report, behavior questions? So a couple of weeks ago my network manager icon stopped appearing; I still get automatically connected to my saved networks and the internet works fine, etc. If I go into System Settings and look at the Network settings I can still see all the networks, only the icon that's normal in the top bar is missing.  I have no idea what caused this, I don't remember installing anything. I think it just happened after a regular update.  I tried these commands to make sure I was properly up-to-date, (which I found on another forum post):  sudo apt-get -f install sudo apt-get update sudo apt-get upgrade   That didn't do it so I found this post:  and I followed the instructions in those answers:  sudo apt-get purge network-manager-gnome &amp;&amp; \ sudo apt-get install network-manager-gnome   And, quoted from an answer there:     Add nm-applet to the Startup Applications (which may be found by   entering "Startup Applications" in the Dash).   I restarted and then saw no Unity interface at all! So I tried those 3 commands above to update again and thankfully that fixed Unity but I still don't have my network manager.  If I try the command nm-applet in a terminal it says:  nm-applet-Message: using fallback from indicator to GtkStatusIcon   So.... I'm pretty much stumped. Any help would be appreciated. Which tube size to prefer if several options apply? I have 40-622 (a.k.a. 28x1.5") tires, and  offers  two different sizes of tubes - their models "17" and "19" - as applicable for that tire size.  Ok, either one would surely fit well. But apart from weight difference (150 vs. 220 grams), is there any reason to prefer one over another? Is it better to have as small &amp; light tube as possible? Or perhaps a bit larger and heavier one would be more durable, as it wouldn't be stretched as much? Err on which side? Which nvidia driver from additional drivers option should I choose? I have NVIDIA Geforece 7025 / nForce 630a integrated gfx card on my UBUNTU 12.04.And I have 4 options in Additional drivers:    NVIDIA accelerated graphics driver (version 173)   NVIDIA accelerated graphics driver (post-release updates)(version current updates)  NVIDIA accelerated graphics driver (version current)[Recommended]  NVIDIA accelerated graphics driver (post-release updates)(version 173 updates)   I am currently using the First option.  My Problem is there I can't update the version 173 to the latest version of drivers using xswat PPA. Whenever I do it and reboot but the NVIDIA xserver sttings always shows me the same version.  Which option would allow me to update my drivers to the current version? A "non-believer" of C# was asking me what the purpose to extension methods was. I explained that you could then add new methods to objects that were already defined, especially when you don't own/control the source to the original object.  He brought up "Why not just add a method to your own class?" We've been going round and round (in a good way). My general response is that it is another tool in the toolbelt, and his response is it is a useless waste of a tool... but I thought I'd get a more "enlightened" answer.  What are some scenarios that you've used extension methods that you couldn't have (or shouldn't have) used a method added on to your own class? How to install updates via command line? When I log into my web server via SSH I see the information:  88 packages can be updated. 80 updates are security updates   I tried apt-get update then apt-get upgrade but each time I log in I still see the message about updates. How do I install them? Is there any hint that the MoM or anywhere else got support in combating voldemort the first time around? Or is everything completely UK-centric?  I can't remember anyone in the second war being specifically sent from europe to help, even though we see international wizards in book 4. But I don't think all the events of the first war are covered in the books.  So are other bodies like the MoM involved in the first war, or was it isolated in the UK? Why can't I change my desktop background? I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this?