Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring.   Ok, so I'm just looking for some confirmation that I'm doing this correctly.  If we suppose $x,y \in R$ Let's consider $(x+y)^2$,  Then $(x+y)^2 = x^2+xy+yx+y^2$ but $x^2=x$ and $y^2=y$   We can also see that for all $x \in R, x=-x$  So, $(x+y)^2= x+xy+yx+y$  Also, by our given  $(x+y)^2=(x+y)$ so   $x+y =x+xy+yx+y$, Solving this algebraicly gives us $-yx=xy$ but since $(-yx)^2=(yx)$,   We have, $yx=xy$, Therefore R is commutative. Does that about wrap it up?
A ring $R$ is a Boolean ring provided that $a^2=a$ for every $a \in R$. How can we show that every Boolean ring is commutative?
Prove that a function is not injective How do I restore an accidentally deleted initrd image? My linear voltage regulator is overheating very fast I'm using a 5 V / 2 A voltage regulator () without a heatsink. I'm testing the circuit with a microcontroller (PIC18FXXXX), a few LEDs and a 1 mA piezzo buzzer. The input voltage is aprox. 24 VDC. After running for a minute, the voltage regulator starts to overheat, meaning it burns my finger if I keep it there for more than a second. Within a few minutes it starts to smell like it's burnt. Is this a normal behavior for this regulator? What could cause it to heat that much?    Other components used in this circuit:  L1:   R2:   F1: By the "No Hair Theorem", three quantities "define" a black hole; Mass, Angular Momentum, and Charge. The first is easy enough to determine, look at the radius of the event horizon and you can use the Schwarzschild formula to compute the mass. Angular Momentum can be found using the cool little ergosphere Penrose "discovered". However, I don't know how to determine the charge of the black hole.   How can an electromagnetic field escape the event horizon of a Reissner-Nordström black hole? Is there any experiment we could theoretically do to a black hole to determine its charge? I know from reading about the gravitational waves detected by Ligo, that when an object has angular acceleration, it produces gravitational waves.  I'm wondering if an object creates gravitational waves when only accelerating in one direction, however?   I'm also curious as to how the nature of the waves would differ in this case. I am trying to solve this exercise from Jech's book on set theory:  Ex. 3.6: There is a well ordering of the class of all finite sequences of ordinals such that for each $\alpha$, the set of all finite sequences in $\omega_{\alpha}$ is an initial segment and its order-type is $\omega_{\alpha}$. Definition: If $W$ is a well ordered set and $u \in W$, then the set $\{ x \in W \mid  x &lt; u \}$ is an initial segment in W  Any help? Groups of order $56$ Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove?  My idea: consider the case $n_2=7,n_7=8$. Then the number of elements in all Sylow $2$ -subgroups and Sylow $7$-subgroups is at most exactly $(7-1)*8+(2-1)*7+1=56$. This cannot give a contradiction. We need to prove that $G$ is simple. Then $|G||(8!/2)$. To prove $G$ simple, we need to prove there does not exist normal subgroups of order $2$ and order $4$. But I do not know how to prove. Executing SSIS 2012 package that has script components from external application Determining bare earth DEM from unclassified LAS file? I have data in LAS format with RGB values created from aerial photogrammetry using a UAV. I am  trying to find a solution to extract the bare earth DEM from the point cloud.   I have tried SAGA, Fusion, MCC-LIDAR, but it is seems they need the LAS file to be already classified (which it naturally isn't). Can anyone point me in the right direction with a brief explanation of the process?   Generally, I would need to process about 100 mill points at a time (can tile them if needed). How can I check if a program exists from a Bash script? When running Internet Explorer from the the modern UI, I can't find out how to access tabs.  I can create tabs using Ctrl + T, but I'm not sure how to switch between tabs, see what tabs I have open, etc.  How do I navigate between tabs when using the modern UI Internet Explorer? Can a well-timed Twisted Image kill a player using Tree of Redemption? What happens if my opponent activates  and I cast  with the life-exchange ability on the stack?  Looking at the comp rules, these seem most relevant:     701.8a A spell or ability may instruct players to exchange something (for example, life totals or    control of two permanents) as part of   its resolution. When such a spell or ability resolves, if the   entire exchange can’t be completed, no part of the exchange occurs.      Example: If a spell attempts to exchange control of two target creatures but one of those              creatures is destroyed before the spell resolves, the spell does nothing to the other              creature.      112.7a Once activated or triggered, an ability exists on the stack independently of its source.    Destruction or removal of the source   after that time won’t affect the ability. Note that some    abilities   cause a source to do something (for example, “Prodigal Sorcerer deals   1 damage to    target creature or player”) rather than the ability   doing anything directly. In these cases, any    activated or triggered   ability that references information about the source because the   effect    needs to be divided checks that information when the ability   is put onto the stack. Otherwise, it    will check that information   when it resolves. In both instances, if the source is no longer in the   zone it’s expected to be in at that time, its last known information   is used. The source can still    perform the action even though it no   longer exists.   Is the entire exchange ability countered because the Tree is dead by the time it resolves? Or do we use the "last known state" (toughness 0), thereby killing its controller? Easiest way to sort DOM nodes? If I have a list like this:  &lt;ul id="mylist"&gt;     &lt;li id="list-item1"&gt;text 1&lt;/li&gt;     &lt;li id="list-item2"&gt;text 2&lt;/li&gt;     &lt;li id="list-item3"&gt;text 3&lt;/li&gt;     &lt;li id="list-item4"&gt;text 4&lt;/li&gt; &lt;/ul&gt;   What's the easiest way to re-arrange the DOM nodes to my preference? (This needs to happen automatically when the page loads, the list-order preference is gained from a cookie)  E.g.  &lt;ul id="mylist"&gt;     &lt;li id="list-item3"&gt;text 3&lt;/li&gt;     &lt;li id="list-item4"&gt;text 4&lt;/li&gt;     &lt;li id="list-item2"&gt;text 2&lt;/li&gt;     &lt;li id="list-item1"&gt;text 1&lt;/li&gt; &lt;/ul&gt; What is the easiest way to export just the selected objects in Illustrator? The title says it all, I want to know what the simplest way to export the selected objects.  In Inkscape, there is a menu option called export selected which does just this.    The aforementioned option lets you export the selection as a composite image or as a batch of individual images.  Both would be nice, but in this scenario I want to export the selected objects as a single image.  I can already create a new document, and paste in the selected items.  I am looking for a way that is easier than this.  How can I save / output / export only the objects I currently have selected, to an image file, in Adobe Illustrator CS5? The message that appears in the reputation tab of a user's profile when they gain or lose reputation due votes disappearing when another user is deleted is simply:     user was removed   This message is confusing and unhelpful. Below are comments that demonstrate this confusion, . I don't have any quotes from the users who could not find it and remained confused.  There is , and . However, the message has not been changed.  I know that in the past the developers have been reluctant to direct users to meta in some situations, such as question bans, because it may cause users to post here appropriately. That does not seem to apply to this case. This is not typically a case of ""; most of us have probably seen the message at some point.  Why is this message vague and unlinked? What negative behaviour does this confusion mitigate?       My first interpretation was that I was the user who was removed!        I just simply wished the explanation on the reputation page was better. Seems like I shouldn't have had to come to the meta section to figure this out.        It must be better to show "User  was removed". For a moment I just thought that I was removed :-(        the information should be better. I lost 10 points yesterday by "user removed" and assumed that I was the one being removed, leaving me wondering why and feeling somewhat embarrassed.         The recalc policy is fine. But the red mark and "user was removed" looks cryptic and unnecessarily negative--especially since the only user name mentioned on a profile is one's own. So the natural impulse is to bind "user" to the account you're looking at. (It was jarring to me and I could tell what it meant, but still this could be clearer.) I agree that orange is better than red and ideally all of these things would have hyperlinks to a help page (or at least a meta post) explaining what the deal is.        +1 (to HostileFork) for the idea of linking to an explanation of 'User was removed'. It just took me 10+ minutes to find this particular thread and read through, when I could have simply clicked through! I would also like a little more info on why user was removed (abuse, friend upvotes, rules, etc) , as that could help educate the community as a whole on what behaviors to report.        I could care less about the rep gain or loss, but when you have a loss it is a scary message without context. I honestly thought for a moment that my account had been inhibited in some way. No, it didn't say "Your account was removed." Still, there is that moment of "what did I do wrong?" which seems silly to cause simply by not posting a few extra words. A clearer "A user who voted on your posts was removed along with reputation changes they caused" would have gone a long way to not cause me to question my participation in the review queues as of late.         My problem is that the message "user has been removed" is not a clickable link like the other messages - so I had to search meta to discover this question to find out what it meant. Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? Detecting Color of iPhone/iPad/iPod touch? Quadratic Formula in Complex Variables Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers. I just came across this code and a few Google searches turn up no explanation of this mysterious (to me) syntax.  Hashtable^ tempHash = gcnew Hashtable(iterators_);  IDictionaryEnumerator^ enumerator = tempHash-&gt;GetEnumerator();   What the heck does the caret mean? (The gcnew is also new to me, and I asked about that .) I need a little hint in a proof on convergence of a sequence. The problem I have is: Suppose $(x_n)$ is a monotone decreasing sequence of real numbers such that $\sum x_n$ converges, prove that:  $$\lim_{n \to \infty} nx_n=0$$  My idea is: proving this fact aims to prove that given $\epsilon &gt;0$ there's some natural $n_0 \in \mathbb{N}$ such that if $n &gt;n_0$ we have $|nx_n|&lt;\epsilon$.  Now, I know that $\sum x_n$ converges, so that given $\epsilon'&gt;0$ there's some $k_0 \in \mathbb{N}$ such that if $k &gt; k_0$ we have:  $$\left|\sum_{i=1}^{k}x_i - S\right|&lt;\epsilon'$$  Where $S = \sum x_n$. Now, since $(x_n)$ is monotone decreasing we know that we must have $x_1 &gt; \cdots &gt; x_k$ so that the sum of all $x_i$ should be less or equal to $k x_k$. So I know that I have:  $$\left|\sum_{i=1}^{k}x_i - S\right|\leq\left|\sum_{i=1}^{k}x_i\right|+|S|\leq|kx_k|+|S|$$  I feel that the proof will come from this, however I'm stuck at this point. Can someone give just a little hint on how to proceed from here?  Thanks very much in advance for your help.