Show that if, for the ring $(A, \bigtriangleup, \bot)$, $\bot^2x=x$ then $(A, \bigtriangleup, \bot)$ is a commutative ring.   Since no elements of $A$ is provided and no operation is given, I am not sure how to proceed with this problem.  I have tried applying the simetric element $x^{\bigtriangleup}$ to the expanded equation:  $$ \bot^2x=x \\ x \bot x \bot x = x $$  But it is not working out. How do I proceed?
Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring.
If someone is my proxy, I am their _____? Bonus question: If I am a proxy for multiple people, what's the collective noun for that group? Kaplansky's theorem of infinitely many right inverses in monoids? 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? Java, Cannot reduce the visibility of the inherited method from object What's the best order to read HP Lovecraft's novels? I've always been interested in the Cthulhu mythos and I wanted to get into HP Lovecraft's books. What is the best book to start with? I know that Cthulhu was first mentioned in Lovecraft's Call of Cthulhu in 1928. Is that the best book to start with? Should I just start reading Lovecraft's books from his first publication onward? How can I prevent Windows from prompting to restart my computer after an update? Show that $f_n(z)=z^n$ converges locally uniformly on $|z|&lt;1$ but not uniformly.   I am really confused. I saw a seemingly reliable reference stating that $z^n$ does not converge uniformly to $0$ on $\{|z|&lt;1\}$ but it does converge uniformly on $\{|z|\le r,r&lt;1\}$. What is the nuance between the two former sets? With $r_n&lt;1,r_n\to 1$, shouldn't one conclude that $f_n(z)$ uniformly converges on $\{|z|&lt;1\}$? And besides, this source tells me what I have to prove for Homework is disproved, which really makes it more frustrating.  I would appreciate any help. The annihilator of a module is an ideal Let $R$ be a ring and let $M$ be an $R$-module. Define the annihilator of $M$ to be the following subset of $R$:  $Ann_R(M) =\{r\in R \mid rm = 0_M  \text{ for all } m\in M \}$  Prove that $Ann_R(M)$ is an ideal of $R$.  please help as much as you can a full answer is greatly appreciated   Thanks I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? What does it mean for a QFT to not be well-defined? It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization group to extrapolate the coupling up to a huge scale, where it formally becomes infinite.  First of all, it seems bizarre to me to use the renormalization group to increase resolution (go to higher energies). The smearing in the RG procedure is irreversible, and the RG doesn't know about the non-universal parts of the theory which should be important at higher energies.   I have read in a note by Weinberg that Kallen was able to show that QED was sick using the spectral representation of the propagator. Essentially, he evaluated a small part of the spectral density and showed that it violated the inequalities for the field strength $Z$ that are imposed by unitarity + Lorentz invariance, but I do not have the reference.  So what I would like to know is :   What does it take to actually establish that a QFT is not well-defined?   In other words, if I write down some arbitrary Lagrangian, what test should I do to determine if the theory actually exists. Are asymptotically free theories the only well-defined ones?   My initial thought was: We would like to obtain QED (or some other ill-defined theory) using the RG on a deformation of some UV CFT. Perhaps there is no CFT with the correct operator content so that we obtain QED from the RG flow. However, I don't really know how to make that precise.   Any help is appreciated. Series where the planet’s air is really polluted and there is a portal to take you to a world where the air is clean The movie was released in the last 15 years, probably between 2008-2018. &quot;Pre-movie&quot; (like what happens before the main part of the movie begins):  The movie was in the future. The people have to wear gas masks to go outside because the air is so polluted. airpollution smog 2 child policy they had 3 kids the guy goes to prison There is some sort of transportation portal that brings some important people 85mio years in the past on another timeline The guy runs to the portal and gets there with his entire family  &quot;Main movie&quot; (most of the movie):  The guy is on this new planet and I think he gets imprisoned. The people built a camp with walls because there are dinosaurs that can easily kill people. Then a few other things happen I don't clearly remember. I think there is also some sort of love story. Then the dinosaurs attack the camp and they have to defend it. If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.  I need to proof this theorem and I have something like this below. However, I do not know how to continue this proof.   $(x+y)^{2}-(x+y)=x^{2}+xy+yx+y^{2}-x-y=(x^{2}-x)+(y^{2}-y)+xy+yx\in Z(R)$  so $xy+yx\in Z(R)$ Host name changed remotely by wifi? Am am currently visiting TU Wien and today I connected my Debian Linux laptop to their eduroam wlan using wpa_supplicant and the credentials of my home institute - as always when I am visiting another scientific institution.  When I opened a terminal I noticed that my command promt was showing a different host name, and in fact, excecuting hostname gave me e244-082.eduroam.tuwien.ac.at instead of the usual host name of my machine x301.  I am very puzzled by this. How on earth can it be possible that connecting to a wlan changes my host name without my consent? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen? How is double negation interpreted in English? Can a PC with the Alert feat alert his friends? Assume the following situation: The party has been surprised by an ambush. Every PC is surprised except PC Bob because he has the Alert feat. Thanks to his high initiative bonus Bob gets to go first. He shouts "look out" to alert his party members of the imminent threat. The idea is that his party members are no longer surprised because he warned them.   While I think that this makes sense from a storytelling perspective, I am afraid this would be too strong. as it allows one PC with the Alert feat to give the "not surprised" benefit to other members of his party pretty much for free.  Is this allowed? Are there rules or an official ruling or similar on this?  If not, do you have any experience with this situation, and how did it play out in your games? When "Questions that may already have your answer" (similar questions) appear as a right column, it overflows and overlaps the footer.  (shown in the figure)        This happens on all SE network sites on which I have asked questions.  Also this happens to other columns also like "How to Format". Who are the diamond moderators, and what is their role? I have the following code that will remove lines with the pattern banana and 2 lines after it:  sed '/banana/I,+2 d' file   So far, so good! But I need it to remove 2 lines before banana, but I can't get it with a “minus sign” or whatever (similar to what grep -v -B2 banana file should do but doesn't):  teresaejunior@localhost ~ &gt; LC_ALL=C sed '-2,/banana/I d' file sed: invalid option -- '2' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,-2 d' file sed: -e expression #1, char 16: unexpected `,' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,2- d' file sed: -e expression #1, char 17: unknown command: `-' (Revised) Prove that $\phi$ is a group homomorphism and find the kernel. Quoting " Let $\phi : \Bbb Z \rightarrow \Bbb Z$ be given by $\phi(n) = 7n$. Prove that $\phi$ is a group homomorphism. Find the kernel, and the image of $\phi$."   My understanding:  Part 1:   Given $n,m \in \Bbb Z$, let's check that $\phi(n+m) =\phi(n)+\phi(m)$  $$\phi(n+m) = 7(n+m) = 7n + 7m = \phi(n)+\phi(m)$$  Therefore as the group operation in $\phi$ is preserved, $\phi$ is a group homomorphism.  Part 2 (revised):   $\phi$ is one-to-one as:  $\space \space \phi (n) = \phi(m) \Rightarrow 7n =7m \Rightarrow n=m  $. We also know that $\phi(e)=e$ by the property of homomorphism. Therefore, no other object than $0$ in the domain can map to $0$ in the codomain. It follows that: $$\ker(\phi)=\{ x \in \Bbb Z : \phi(x)=e\}=\{e\}$$   Part 3 (revised):  I claim that the only possible image-elements of $\phi$ are the multiples of $7$ in $\mathbb Z$ denoted $7\mathbb Z$.   To prove this I have to show that Im$(\phi) \subset 7 \Bbb Z$ and that $ 7 \Bbb Z \subset $ Im$(\phi)$.  Proving Im$(\phi) \subset 7 \Bbb Z$ :      $\forall y \in $ Im$(\phi)$ such that $  y =\phi(x)$ where $x \in 7 \Bbb Z$. It follows that $y \in 7 \Bbb Z$, therefore Im$(\phi) \subset 7 \Bbb Z$   Proving $ 7 \Bbb Z \subset $ Im$(\phi)$:   $\forall y \in  7\Bbb Z$ such that $  y =7x$ where $x \in \phi(x)$. It follows that $y \in \phi(x)$, therefore $ 7 \Bbb Z \subset $ Im$(\phi)$.   Any input to my understanding is much appreciated. What Lagrangian counterterms might be needed in a $\phi^ 6$ theory in 3D?