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Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective.  Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker f=R$ then $R/R \cong \{0\}$, but $T \neq \{0\}$, which is a contradiction. I can't see why does the last part means contradiction.   Thanks for any assistance! $\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums? How can I permanently enable cheating? I'm giving Deus Ex a quick playthrough as a refresher before jumping into Human Revolution. In the interest of expediency, I've decided to cheat. ^.^  Normally, cheats must be re-enabled for each level when playing the original DX. I've encountered instructions on several sites (word-for-word identical instructions, so clearly copy/pasted) for permanently enabling cheats, but they aren't working for me:     Because you will need to redo this for every level, you can open up 'user.ini' and add 'bCheatsEnabled True' in the [DeusEx.DeusExGameInfo] section of the file to allow cheats to be activated forever.   Not only is that not working for me, it doesn't even look right compared to the rest of the file (space instead of =). With increasing desperation, I've tried adding that line (both verbatim and switching to an equal sign) to every section of both User.ini and DeusEx.ini which looks even remotely plausible — even creating a [DeusEx.JCDentonMale] section, just in case.  Does anyone know how to get this working? I'm running DX:GOTY version 1.112fm from Steam on Windows 7. The formula for a one sample t-test, as given by  is  $t = \dfrac{\bar{x} - \mu_0}{s\big/\sqrt{n}}$  However, in this equation, if the sample standard deviation, $s$, is zero, $t$ will be undefined. Is there any reasonable way to get a $t$ value in this case? When you send a group text message in iOS, it appears as if you automatically opt in to receive any and all replies to that message. Is there a way to avoid being included in group replies? Is there a way to remove oneself from a group chat? proper use google webmaster tools URL Parameters There are some urls like http://domain.com/user/:username and http://domain.com/post/:id on my website, the username and id is the parameter from database. I try to let google crawler more efficiently , can I use google webmaster tools &gt; URL Parameters &gt; add parameter ? I read the document but still don't get it how to set up? the column Parameter and the rest what should I filled? or this tools only for get method parameter? QM and Renormalization (layman) I was reading Michio Kaku's Beyond Einstein. In it, I think, he explains that when physicsts treat a particle as a geometric point they end up with infinity when calculating the strength of the particle's field as you approach the particle.  First, did I get that part right?  Second, how does (in very simple terms) renormalization try to fix this in Quantum Mechanics? If you can include some type of word picture that would be great. Accepted answer will go to the clearest explanation.  Update  I got a little further in the book (lol) and Kaku talks about using symmetry to remove the divergences in the math. I'd appreciate an answer that incorporates this also. Thanks guys! “community wiki” checkbox in new question shows a dialog twice when self-answering Click Answer your own question Check community wiki A dialog box     Are you sure you want to make this post Community Wiki?   will appear, Click OK or Cancel; The same dialog will appear again.   Google Chrome 44.0.2403.107 / Windows8.1 / site rev 2015.7.25.3609 Is there an AES identity key? The following questions are of pure theoretical nature. I don't have an application in mind.   Is there a key that makes AES the identity function? Is there a key that makes AES the identity function for certain inputs? Is it known whether such keys might exist? Are such keys known for other ciphers?   For example, I want to encrypt "Hello StackExchg" with AES with said key and the ciphertext should be again "Hello StackExchg".  When I encrypt "Hello StackExchg" with the perfect cipher one-time-pad, there is a possibility that the resulting ciphertext is again "Hello StackExchg". This possibility must exist (though it is unrealistically small), otherwise, the ciphertext would not be perfectly random but depending on the plaintext. The corresponding key for one-time-pad would be all zeros. Is there any way to view saved wifi passwords on the iPhone? Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field 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? How can I exclude API keys, passwords, and so on from configuration management? I am using Drupal 8 and drush cex to commit my configuration to my repository.  However, I want to keep passwords and API keys out of the repository.  How do I do that? (selectively exclude sensitive information from configuration management).  For example, I'm using the  module, but I noticed that in addition to the SMTP server and sender's name, etc., it also exports the API key to code, which should not be exposed.  I want to manage module configuration with configuration management, but I don't want to commit the key. Why would layout_width="wrap_content" take up the entire width of the screen? 17.10 Super key to open applications I've upgraded my laptop from 17.04 to 17.10 and the super key no longer brings up the applications menu. Does anyone have any idea how I can change this?   Shortcuts menu doesn't let me change Super + A to just Super.  Edit: I'm running the standard desktop version of Ubuntu 17.10 I am looking for a injective cochain map $\psi: C^*\rightarrow D^*$ such that the map $\Psi^i$ from $C^i$ to $D^i$ is injective, but the map $\psi^*$ from  $H^i(C^*)$ to $H^i(D^*)$ is not injective for any $i\geq0$. Spacelike interval and length contraction I am having a serious trouble finding an agreement between the two facts in Special Relativity:  (1) given two events $A$ and $B$ in spacetime separeted by a spacelike interval $s$, the distance in space between the two events will be the shortest in the reference frame which the two events occur simultaneously:   The invariance of the interval says $s^2=(\Delta x)^2-(c\Delta t)^2$ and $\Delta x$ is minimum when $\Delta t=0$  (2) the length contraction says $L=\dfrac{L_0}{\gamma}$ where $\gamma=(1-(v/c)^2)^{-1/2}$ , $L_0$ is the proper length of an object in a reference frame which the object is at rest and $L$ is the length of the object in a reference frame which the object moves with speed $v$ in the $x$ direction.  This says that the lenght of the object in the $x$ direction is the largest in the comoving reference frame of this object.  The question is: According to (1), the reference frame in which two events $A$ and $B$ occur simultaneously is the reference frame that gives the shortest distance in space between the two events. Let $A$ and $B$ be the two end points of a meter stick in a reference frame in which the stick is at rest and at time $t=0$. This two events occur simultaneously, so the distance in space between the two is the shortest acording to (1). But according to (2) shouldn't this distance be the largest?  What is wrong? What are metaclasses in Python? Difference between static class and singleton pattern? I need to use a different version of circuitikz. How can I do that?