Let $R$ be a finite Boolean ring with $1 \neq 0$. Show that $R\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$.   This is exercise 2 on p267 in the book abstract algebra of Dummit and Foote. The author gives the hint to use that if $e$ is an idempotent of a ring $R$ with $1$, then  $$R \cong Re \times R(1-e)$$  where $Re$ is a ring with identity $e$ and $R(1-e)$ a ring with identity $1-e$.  My attempt:  If $R = \{0,1\}$, then $R \cong \mathbb{Z}_2$ is trivial.  Hence, we may assume that $\{0,1\}$ is properly contained in $R$. Thus,we can pick an element $x \notin\{0,1\}$. We obtain because every element in a Boolean ring is an idempotent:  $$R \cong Rx \times R(1-x)$$  $Rx$ is a ring with identity $x \neq 0$ and $R(1-x)$ is a ring with identity $1-x \neq 0$. Moreover, $Rx$ and $R(1-e)$ have cardinality strictly smaller than the cardinality of $R$. These rings are also Boolean. Induction yields the result. $\quad \square$  Is this correct?
I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook.
Processing time/speed for Mosaic to New Raster I have a question on speeding up “Mosaic to New Raster” or finding an alternative.  After completing a suitability analysis we are trying to define movement corridors.  We used a modified CircuitScape that left 4 folders, each containing 1080 raster tiles across our study area.  We now need combine all 1080 rasters in each folder then combine these 4 combined rasters into a final raster which will be analyzed for paths.  I tried a python script that just wouldn’t work so I went on to Model Builder.  It iterates through the grids in the folder, collects the values (raster names), then mosaics all to a new raster.  It has been running for 5 days now and the results window indicates it is working.  I’m hoping it will conclude in the next day or 2.  But, we still have 3 more folders, each with 1080 rasters, that need to be mosaicked.  So….is there a way to speed up Mosaic to New Raster or is there an alternative I could use to produce the final grid?    The cell size is 270 meters, 540 x 540 columns and rows, 1 band, 32 bit floating. How do I get jQuery to select elements with a . (period) in their ID? Given the following classes and controller action method:  public School {   public Int32 ID { get; set; }   publig String Name { get; set; }   public Address Address { get; set; } }  public class Address {   public string Street1 { get; set; }   public string City { get; set; }   public String ZipCode { get; set; }   public String State { get; set; }   public String Country { get; set; } }  [Authorize(Roles = "SchoolEditor")] [AcceptVerbs(HttpVerbs.Post)] public SchoolResponse Edit(Int32 id, FormCollection form) {   School school = GetSchoolFromRepository(id);    UpdateModel(school, form);    return new SchoolResponse() { School = school }; }   And the following form:  &lt;form method="post"&gt;   School: &lt;%= Html.TextBox("Name") %&gt;&lt;br /&gt;   Street: &lt;%= Html.TextBox("Address.Street") %&gt;&lt;br /&gt;   City:  &lt;%= Html.TextBox("Address.City") %&gt;&lt;br /&gt;   Zip Code: &lt;%= Html.TextBox("Address.ZipCode") %&gt;&lt;br /&gt;   Sate: &lt;select id="Address.State"&gt;&lt;/select&gt;&lt;br /&gt;   Country: &lt;select id="Address.Country"&gt;&lt;/select&gt;&lt;br /&gt; &lt;/form&gt;   I am able to update both the School instance and the Address member of the school.  This is quite nice!  Thank you ASP.NET MVC team!  However, how do I use jQuery to select the drop down list so that I can pre-fill it?  I realize that I could do this server side but there will be other dynamic elements on the page that affect the list.  The following is what I have so far, and it does not work as the selectors don't seem to match the IDs:  $(function() {   $.getJSON("/Location/GetCountryList", null, function(data) {     $("#Address.Country").fillSelect(data);   });   $("#Address.Country").change(function() {     $.getJSON("/Location/GetRegionsForCountry", { country: $(this).val() }, function(data) {       $("#Address.State").fillSelect(data);     });   }); }); (How) Can SE sites change their reputation payout? A recent question on  reminded me that there is a  about awarding 10 rep per question upvote. This is mainly due to the relative difficulty and usefulness of asking/answering being different on the site (or appearing to be, at least).  The highly voted answer on PPCG is "Yes, we should do that," but it's been well over three years since it was posted. The answer there said that (at least at that time) meta.so had this implemented already, so it wouldn't/shouldn't be hard to do.  Given that it's been quite some time, I don't know if that's (still) true. I know that meat has undergone changes since then, but as long as I've been actively participating, meta.se (and meta.so before the split) has given 5 points per question upvote, not 10.  My question is: What would the process to change it look like? It's been discussed on meta.ppcg and agreed upon, so what's the next step? Or is it not feasible at all? Converting string to tuple and adding to tuple Changing local IP address with networksetup command on OSX? My computer routinely receives connection from another host in the local network. However, the router changes its ip address quite frequently, and maybe netsh's equivalent, networksetup, can help? How is evolution possible in contemporary humans? I would like to know if evolution is continuing to happen in modern humans, assuming things like existence of the nuclear family structure, fidelity to one partner, etc. It seems to me the answer would be NO because evolution depends on differential reproductive rates, but in the modern world, all male humans have roughly 2.5 (or whatever the number) kids. Add in the process of culturally modified selection pressure, and it seems to me that even an "unfit" male would end up having a couple of offspring. The fittest male (or female) is no better off than his or her contemporaries because of this "leveling" effect.   However, the impression I get from the popular science media is that scientists think evolution is continuing to happen. I would like to know what the actual scientific consensus is, and why. Thanks. Given the sentence:     Most people who live and work near Washington, DC, would have trouble      imagining dinosaurs walking around the area.   Grammatically, what would ‘imaging’ be called in this case? Present participle or gerund or something?  I mean, if it is a present participle, ‘imagining~’ might be a description of the subject. And if it is a gerund, ‘imagining~’ might be an object of the verb phrase of ‘have trouble’. How do they explain this ‘imagining’ after ‘have trouble’? Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help. Maven project with native dependency and copying files Show that $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot 2^{\aleph_1}$ So basically I want to show:  $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot2^{\aleph_1}$ , Where $\omega\leq\alpha&lt;\omega_1$  There is a proof that I found;    But I don't understand his proof for the limit case of $\alpha$.  To be more specific, I dont understand why $ \left( \sup_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1} \le \left( \prod_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1}$  Any insights or help is deeply appreciated.  Cheers Mapping building to only one grid which contains more than half of its area using QGIS? Where is the rest of Yoda's species? We never seem to see any beings of Yoda's species - except, of course, Yoda himself.  This leads us to the question of where the rest of the li'l critt'rs are. Yoda had to have parents, correct? So what happened to them? Was Yoda, put simply, just one of the last of a dying species? Shortcut in Windows 7 to switch between same application's windows, like Cmd + ` in OS X Is there a shortcut in Windows 7 to switch between windows in a given application only? I very much like this feature in OS X; what is its equivalent in Windows? I am a beginning indie developer, and I want to get the word out about my game. What sites can I go to to ask about reviews of my game, and where might I be able to talk about and get feedback on my game from players? line of mathematicians guess their own hat color out of c colors There is a common problem in which a long line of N mathematicians are each given a hat that is either red or blue.  They cannot see their own hat but can see all in front of time and can hear any response. They must state their own hat color from the back of the line forwards and no information may be passed besides their guess.  They may have a planning session in advance.  What is the maximum number of correct guesses can they guarantee?  The answer is (look away if you don't know) $N-1$.  If, instead of 2 colored hats, we use $c$ different colored hats, how many can we guarantee?  Assuming $N&gt;&gt;c$  I believe the answer to be $N-c+1$ and have confirmed this for $c=2, 3$, and $4$.  I believe this should be simple to prove by using the concept of "$x$ formulas allow you to solve for $x$ variables" but I am not well versed enough in pure mathematics to answer this. How to open a terminal from within Atom? I'm currently working with Atom, and sometimes I want to open a terminal on the directory my working file is. To do this, I usually have to open the folder in nautilus (right clicking the file in atom) and then I open a terminal on that directory from nautilus (right click).  I wonder if there is a way to open the terminal in the file directory without having to open nautilus first. dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with different file `/usr/lib/xorg/protocol-precise.txt', not allowed I just installed Ubuntu (12.04.4) and I upgraded to 13.10  I have no idea how to fix this :/  $ sudo apt-get install -f  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   xserver-common-lts-saucy 0 upgraded, 0 newly installed, 1 to remove and 2 not upgraded. 1 not fully installed or removed. After this operation, 1,718 kB disk space will be freed. Do you want to continue [Y/n]? y (Reading database ... 175174 files and directories currently installed.) Removing xserver-common-lts-saucy ... Removing 'diversion of /usr/lib/xorg/protocol.txt to /usr/lib/xorg/protocol-precise.txt by xserver-common-lts-saucy' dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with   different file `/usr/lib/xorg/protocol-precise.txt', not allowed dpkg: error processing xserver-common-lts-saucy (--remove):  subprocess installed post-removal script returned error exit status 2 Errors were encountered while processing:  xserver-common-lts-saucy E: Sub-process /usr/bin/dpkg returned an error code (1) How to download/read only the first 80KB of a file? Entity Query field condition for Webform submissions I'm trying to query webform submissions based on a field value. The field is called "ticket_for". Somethings I tried and do not work:      -&gt;condition('ticket_for',$adnummer)     -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS') ...   What kind of condition should I be using in a query like this?  $query = \Drupal::entityQuery('webform_submission') -&gt;condition('webform_id', 'ticket') -&gt;condition('entity_id', $eventID) -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS'); Is there a place where I can look what updates I've installed?