For a ring $R$, I must show that if R has a left identity that is not a right identity, then there is more than one left identity.   Could I do this by contradiction, using 2x2 matrices and show that a given set has two left identities that are not right identities?
There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct.  To see they're distinct, I suppose $v+(1-vu)u^n=v+(1-vu)u^m$ for distinct $n$ and $m$. I suppose $n&gt;m$. Since $u$ is cancellable on the right, this implies $(1-vu)u^{n-m}=1-vu$. Then $(1-vu)u^{n-m-1}u+vu=((1-vu)u^{n-m-1}+v)u=1$, so $u$ has a left inverse, but then $u$ would be a unit, and hence have only one right inverse.  Does the same theorem hold in monoids, or is there some counterexample?
Say I have a metric space $(X, d)$ and a set $A\in X$.  I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help? Creating new users using PHP not recognized by Apache 2.4 I have a server setup where new linux users are created using (PHP)  shell_exec ( '/usr/sbin/useradd ' . $username ); exec ('usermod -a -G ' . $username . ' '. $username); #add user into the usergroup exec ('usermod -a -G ' . $username . ' www-data'); #add usergroup with the www-data   And then the user script changes the owner and group of the files to this owner. The server is setup with suPHP which then allows these files to be run as the owner.  This setup worked fine with Apache 2.2 but is breaking in Apache 2.4. Everything works fine except when I try to oepn the files, it gives a 403 forbidden error. Restarting apache after the user is added fixes the issue but that is not a viable solution for me. The users are added using a web server and hence I need to be able to do this without having to reload/restart apache everytime.  I have searched for serveral hours and tried to fix the issue but haven't been able to do so! I am running ubuntu 14.04 and I am not able to downgrade to apache 2.2 as well. On Linux can I keep a program running in background even after I disconnect? Do mindless creatures get feats? Do creatures that have no Intelligence Score ("Int —" in the stat block) gain Feats? I'm creating some monsters but I failed to find the answer for this question. Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$. Let $f$ be differentiable on $\Bbb{R}$ and let $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$. Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.  Attempt: If $f$ is constant we are done. Otherwise, $f$ is not constant. Suppose there is no such $a$. Then, in particular, $f$ has no extremum $\implies$ $f$ has no point where is goes from increasing to decreasing or from decreasing to increasing. $f$ is continuous since it is differentiable. Hence, since $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$, $f$ must be constant. A contradiction. Therefore either $f$ is constant and $f'(x)=0$ everywhere, or $f$ has an extremum at $x=a$ and $f'(a)=0$.   I feel like my attempt is not formal enough, any correction?  I would appreciate your help. Why lipophilic molecules can pass phospholipid bilayer, in spite of 2 hydrophilic layers? It is commonly told that, hydrophobic/ lipophilic/ nonpolar molecules can quite easily pass phospholipid bilayer, and hydrophilic (polar or ionic) molecules can't pass (when no protein aid that); because hydrophobic nature of the lipid.  But in the same logic, hydrophobic molecules shouldn't pass through the bilayer. Because there are 2 hydrophilic layers in the membrane. i.e.      A and A' .  Then how the hydrophobic molecules can pass through A and A'?   I guess, it is due to thin diameter of A and A' . Is that? Could it be that Goldbach conjecture is undecidable? The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach conjecture is an undecidable proposition. Find the equation of the circle passing through two points (4,4) and (-2.-2) with radius 4 I am trying to solve this question but i cannot seem to find the center (h,k). I know i have to use the equation of circle form '(x-h)^2+(x-k)^2=r^2' and after that form to solution of the 2 point given and simultaneous equations them. I got the answer for that which is h+k=2 and that is where i got stuck. Please anyone help me with this. i also post my working as well. Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is divisible by $n$.  I don't know even how to begin. I really need help. Thanks Is everything in the physical world composite? In philosophy there is a principle that anything composite cannot have existed eternally, since it is preceded by its parts and whatever forces assembled it.  Is everything in the physical world composite in one sense or another or is there something not composite? @Id @GeneratedValue but set own ID value sum of polynoms of given property I was trying to make python version 3 as a default version of Ubuntu,  I used nano ~/.bashrc and added the first line of this answer.    Then It just messed up my terminal.  Now when I am clicking on my Terminal or trying to access by ctrl+alt+t, the terminal does open/run. I can only talk to ubuntu using my terminal how could I fix this?  Please advise!  Pls, note that the terminal icon is present, but the terminal does not open when clicked on. How do I get the terminal to open and work?  Update: I deleted that line from .bashrc. However, I still have a problem opening my terminal. I also see this sign on my ubuntu as well.     saying: a problem occurred when checking for the updates ubuntu 18.04 I have encountered this question while tring to figure out why every sequential space $X$, is a quotient spaces of a metric space. If I understand correctly, given a sequential space, every sequence (including it's limit) can be identified with the space $Y=\{0\} \cup \{\frac1{n+1}|n \in N\}$. So The space $X$ can viewed as a quotient space of $\bigoplus_{(x_n) \in C} \{(x_n) \} \times Y$ where $C$ is the set of all converges sequences in $X$. My question is, How can I show that this space is metrizable?  Thank you! Shir How can I manually create a pixel perfect image sprite? What is your simplest explanation of string theory? It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. How do you close windows/tabs in 2.8? I've tried a lot of stuff, but always end up opening up more. Installing from source - do I need to keep the extracted tarball directory If I'm installing from source, do I need to keep the extracted tarball directory?  So if I download the git tarball. I then do:  tar -xvzf git.tar.gz   This will create a git.x.x. directory, into which I cd, then run ./configure etc. Once I'm done with this process, and git, or whatever is installed, do I need to keep the original git.x.x extracted directory, or is that just used for compiling the program?  I'm somewhat confused by all the directories, and folders used for programs. My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540)