Let $R$ be a ring with unit and let $M(R)$ be a collection of all non invertible elements in $R$. If $M(R)$ is a ideal in $R$, prove that this is the only and maximal ideal  My thoughts: suppose there is a bigger ideal choose some invertible element in the bigger ideal so the bigger ideal is the whole ring because there are two elements such that $u\cdot v=1$.
My question is how would I go about proving this?     Prove that $R$ is a local ring if and only if all elements of R that are not units form an ideal.   I understand that I need to prove both directions so:  $(\Rightarrow)$ Local ring means has a unique maximal ideal, so I want to show that this implies the elements are not units.  $(\Leftarrow)$ non unit elements of $R$ form an ideal, so if I show this is unique maximal ideal I can then conclude local ring?  Any hints would be appreciated.
How to invite friends to Stack Overflow, Server Fault and Super User? Since we're all into this OpenID thing, I think it would be really useful (and quite easy) to implement a friend invitation feature that grabs a list of friends from one's webmail contact list with an option to choose who to invite to SO, SF or SU. Something in the line of Facebook's friend invitation and recommendation feature. I am new to Ubuntu and want to install VMWare Player. What is the step by step procedure to install that? Is there a definition of  and proper explanation of it?  It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter."  How is it though that matter can get charged?  Defining charge as the property of feeling a force with other charged matter seems circular.  What is charge?  Is there a non-circular definition / explanation? Invalid token 'while' in class, struct, or interface member declaration in very simple code I am not sure what the problem is but I keep receiving this error when I try to use a while statement in my code.     Invalid token 'while' in class,   struct, or interface member   declaration   I want to use a while loop to have something continuously update while a statement is true.  The rest of my code is rather long but whenever I type in the syntax:  while(a&lt;b) { //do whatever i want it to do here }   It gives me that compiler error right off the bat. Not quite sure what the problem is. I am doing this in a C# windows application under the Form1.cs file with all the other event handlers (for buttons and such).  Thanks!    I was unaware that loops had to be placed within a method (fairly new to c#), but I tried it and no errors were returned. Thanks for your help everybody!  Previously, I just had the loop within the main class of the program. Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. Can I configure it to allow:   select right-click to copy right-click or ctrl-v to paste? How to make the taskbar blink my application like Messenger does when a new message arrive? Understanding d-separation theory in causal Bayesian networks I am trying to understand the d-Separation logic in Causal Bayesian Networks. I know how the algorithm works, but I don't exactly understand why the "flow of information" works as stated in the algorithm.    For example in the graph above, lets think that we are only given X and no other variable has been observed. Then according to the rules of d-separation, the information flow from X to D:   X influences A, which is $P(A)\neq P(A|X)$. This is OK, since A causes X and if we know about the effect X, this affects our belief about the cause A. Information flows. X influences B,which is $P(B)\neq P(B|X)$. This is OK, since A has been changed by our knowledge about X, the change at A can influence our beliefs about its cause, B, as well. X influences C,which is $P(C)\neq P(C|X)$. This is OK because we know that B is biased by our knowledge about its indirect effect, X, and since B is biased by X, this will influence B's all direct and indirect effects. C is a direct effect of B and it is influenced by our knowledge about X.   Well, up to this point, everything is OK for me since the flow of the information occurs according to intuitive cause-effect relationships. But I don't get the special behavior of so called "V-structures" or "Colliders" in this scheme. According to the d-Separation theory, B and D are the common causes of C in the graph above and it says that if we did not observe C or any of its descendants, the flow information from X is blocked at C. Well, OK, but my question is why?  From the three steps above, started from X, we saw that C is influenced by our knowledge about X and the information flow occurred according to the cause-effect relationship. The d-Separation theory says that we cannot go from C to D since C is not observed. But I think that since we know that C is biased and D is a cause of C, D should be affected as well while the theory says the opposite. I am clearly missing something in my thinking pattern but can't see what it is.  So I need an explanation of why the flow of information blocked at C, if C is not observed. This is relating to the movie. Frodo finally relinquishes his desire for the ring, and it melts in the fires of Mount Doom. The next scene shows the eruption of Mount Doom and the destruction of Barad Dur. Gandalf looks up with tears in his eyes.  I have read the books, and can't seem to remember this being explained or described. I'm just wondering why they depicted Gandalf as being so emotional about the event, when throughout the story it depicts Gandalf as having unparalleled wisdom, faith and knowledge. Out of all the Ainur he listened to Eru's song the longest and contemplated it's meaning.   It strikes me that Tolkien portrayed Gandalf as all knowing, and has ultimate faith that Frodo would carry out the task at hand. So why was he so emotionally surprised that it happened? Especially when he had seen far more surprising things happening throughout the War of the Ring and in the movies he was depicted as emotionally immovable, an almost cold hearted awareness of the evil that lurks within the world.   I understand this is probably going to be down-voted as it has alot of opinion based statements. The TL:DR is: Was Gandalf described in the book as being so emotional at Sauron's fall? Or was this simply for effect in the screenplay? Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. After reading some articles on blender speed, I see that using CUDA enabled GPU computing will be of help in rendering.   I am trying to stitch a shirt on human .obj file. For that I bake 60 frames for it to fall on the human correctly. It takes about 20 seconds for the bake. I'd want to host it real time.  youtube link of the same :   Can I use GPU computing to reduce the bake time just like rendering time ? Calculating latitude and longitude of points using QGIS How do I easily calculate the lat/long of points using the QGIS Field Calculator? I saw a previous related question (), but it deals with a more advanced approach. In ArcGIS Desktop this is similar to the Calculate Geometry function; I just can't find where do this in QGIS. Three-columns text with figures of 2\columnwidth Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$? How do I programmatically create a new content type? Looking for succinct way of describing the act of solidifying an idea into reality Looking for a succinct way of describing the act of solidifying an idea into reality. "Building [idea] out" seems too generic but I'm trying to avoid a long winded, step by step explanation of the process.  Any ideas would be greatly appreciated. Thanks! Circuitikz symbols in TikZ circuit library Some time ago, I asked about . wh1t3 wrote: "However, as far as I know there is nothing the circuitikz package does that the TikZ circuit library can't."  So if it is true that the tikz library is more powerful, how do I get all the symbols described in the  using the TikZ circuit library?  Especially I am missing the symbols for ammeters and voltmeters. Photoelectric Effect - Dependence of current on frequency We all know that the amount of current flowing b/w the plates is independent of the frequency. If I were to ask why doesn't it depend then you'd probably say that it depends on the number of electrons rather than the frequency or you might show me a current vs frequency graph.  But here is my theoretical argument. Electric current is defined as the amount of charge flowing through a point/space/wire per unit time. Let I be current due to photo electrons ejected by photons of some frequency. Now if I would increase the frequency, the kinetic energy of the ejected electron would increases, therefore it will move faster and therefore contribute more current (charge will through the same point many times per unit time) . Thereby, giving a net current greater than I.  What is wrong with my argument?  After thinking a lot, I came up with a possible reason. However fast the electrons may move, but when they hit the metal plate, they would lose their velocity and start moving in drift velocity. Now there is a problem with this argument. If this were to be true then I should expect to see a big chunk of extra electrons getting stuck at the anode.  If my reasoning was correct, then why can't I see a big charge buildup at the anode. Lagrangian and Hamiltonian EOM with dissipative force I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the Lagrangian  $$ L=\frac{1}{2}m\dot{q}^{2}-\frac{1}{2}kq^{2}+f(t)q$$ with Rayleigh dissipation function as   $$ D=\frac{1}{2}\beta\dot{q}^{2}$$ to put in Lagrangian EOM $$0 = \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) - \frac {\partial L}{\partial q_j} + \frac {\partial D}{\partial \dot{q}_j}. $$  On Legendre transform of $L$, I get $$H=\frac{1}{2m}{p}^{2}+\frac{1}{2}kq^{2}-f(t)q.$$  How do I include the dissipative term to get the correct EOM from the Hamiltonian's EOM? I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.