$ \mathbf{Z}[x]$ is not PID.   we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.   The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$.
Is there a symbol that looks like the element sign $\in $ but instead of being curved, looks more like a less than sign $&lt; $? I have trawled through hundreds of symbols and tried detexify. Let $R$ be a commutative ring with identity, and $S$ a multiplicative subset of $R$ not containing $0$. Which step in my reasoning is wrong?   Every element of $S$ is a unit in $S^{-1}R$. Every element of $S$ that's a zero divisor in $R$ is a zero divisor in $S^{-1}R$. But zero divisors can't be units. Therefore it's not possible to localize $R$ at $S$. Irrationality proofs not by contradiction Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction.  When thinking about it, it seems like the definition of irrationality itself demands proofs by contradiction. An irrational number is a number that is not a rational number. It seems then that if we were to find direct irrationality proofs, this would rely on some equivalent definition of irrational numbers, not involving rational numbers themselves.   Are there any irrationality proofs not using contradiction? Why does a neon lamp indicator on a power strip switch flicker in the dark? Neon lamp indicators embedded in power strip switches indicate whether the switch is ON or OFF. When the neon lamp is ON in a dark environment (lights of the room are off), the lamp begins flickering and it sometimes goes off for a few seconds. When I switch the lights of the room on, it stops flickering and remains bright. Even when I unlock my phone in the dark near the lamp, it stops flickering! Is the light-making-mechanism in neon lamps affected by the light intensity of the surrounding environment?  UPDATE: I captured my observations with my iPhone and uploaded it to YouTube. Here is the link of the video:  All the lights of my room were turned off. There were no plugs connected to the power strip. The only source of light was the screen of my MacBook Pro, which was not plugged in.  I increased and decreased the brightness of the computer screen during the video.  I then produced two additional light environments using my phone. Is there any way to make a method with more than one instance? How to create SHA512 password hashes on command line In Linux I can create a SHA1 password hash using sha1pass mypassword. Is there a similar command line tool which lets me create sha512 hashes? Same question for Bcrypt and PBKDF2. Short story (+radio play?) in which astronauts outside their ship slowly drift apart in space Looking for a short story in which the crew of a spaceship for some reason find themselves outside the ship, drifting apart in outer space. They can communicate via their radios but are individually drifting apart.  One by one, they lose communication as they drift further away from each other.  Might have also been adapted into a radio play? According to ,  The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.  I am aware of Russell's Paradox, which explains why not everything is a set, but how can we show the collection of all groups is a proper class? setfacl incorrectly changes group permissions Word for obscuring bad or immoral acts with verbiage I have been in many scenarios where people try to obscure/reduce the intensity of a seemingly bad or immoral act using grammar. E.g. Assuming Thomas killed a dog, you could hear something like:     There's essentially nothing wrong about palliating the miserable life   of a despondent being. If there was any issue here, I'd say it's not   consulting a veterinary doctor first but I've come to understand that some people are just straightforward.   How does one react to the above mumbo jumbo? I think there are similar situations also found in law courts. What can this be referred to as? Grammatification, obscure tactics? Reducing voltage with resistors Prove Or Disprove: tr(AB)=tr(A)*tr(B) $\mathrm{tr}(AB)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_{ij}*b_{ji}$  $\mathrm{tr}(A)*\mathrm{tr}(B)=\sum\limits_{i=1}^n a_{ii}*\sum\limits_{i=1}^n b_{ii}$  Therefore $\mathrm{tr}(AB) \neq \mathrm{tr}(A)*\mathrm{tr}(B)$  Is the proof valid? Why is quadratic integer ring defined in that way? Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation}    \mathcal{O}=\begin{cases}       \mathbb{Z}[\sqrt{D}] &amp; \text{if}\ D=2,3\ \pmod 4\\       \mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]\  &amp; \text{if}\ D=1\pmod 4    \end{cases} \end{equation} where $D$ is square-free.   I understand $\mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]$ is not closed under multiplication if $D=2,3\pmod 4$. But still, isn't it more natural to define $\mathcal{O}=\mathbb{Z}[\sqrt{D}]$ for all square-free $D$? (in that case, it really seems like 'quadratic integer') I wonder what is the motivation of this definition. How do I invoke a Java method when given the method name as a string? If I have two variables:  Object obj; String methodName = "getName";   Without knowing the class of obj, how can I call the method identified by methodName on it?  The method being called has no parameters, and a String return value. It's a getter for a Java bean. Texture Channel animated, doesn't appear in viewport I'm creating an educational video, and for efficiency, I'm using a Cyclic repeater to repeat an on-page/off-page effect.  Every loop, I have the texture channel switch to the next image, rather than duplicating all the objects/animations. This works properly and appears when rendering the animation, but does not appear in the viewport. It sticks to the original UV image that was mapped.  Is there a way to have this change appear in the viewport? Is there any point in taking the alternate exits in Bit Trip Runner 2?  I can't see any benefit to doing so in terms of unlocking a different path in the overworld, so is it just for your own benefit? Can I mix MySQL APIs in PHP? Should you use "who" or "that" when talking about multiple people doing something? Which of the following is correct?        There were 10 people that went to the store.   There were 10 people who went to the store.      Edit:  Which of the following is correct?        There were 10 people that had brown hair.   There were 10 people who had brown hair. We're running a couple SQL servers in Azure that are set up with an Always On availability group and Windows Failover Clustering. The servers are Windows 2019 and we're running SQL Server 2019. When we set up the cluster, it was set up to use a Distributed Network Name instead of a static cluster IP address. Thanks to this we shouldn't need an internal load balancer according to these notes: .  I'm struggling to understand exactly how this works though. Based on what I read, it seems like our connection strings will point to the DNS name of the cluster (let's call it AgCluster). If I look in DNS, there is an A record for AgCluster pointing to sql1 and another pointing to sql2. When I use AgCluster in my connection string it seems to always connect me to the primary server, even if I have ApplicationIntent=ReadOnly set. When I query @@SERVERNAME I always get the same server.  So with the Distributed Network Name setup, what should I use in my connection strings to make sure read/write queries go to the primary and read only go to a secondary? Any guides on setting this up in general would be helpful. Thanks! I'm using Android 4.4.2 (on ). I've a Windows PC at home which has highspeed broadband internet connection. I don't have wireless router or access point. I just want to use PC's internet connection on my phone. I've tried multiple methods but none seems to work.  Method 1: Using native Android 4.4.2 option  In my phone when I connect my phone, in Settings &gt; Wireless &amp; networks &gt; USB Internet (sharing Windows PC internet via USB cable) checkbox, which I checked then it asked me to choose my Windows version (which is Windows 8.1 but since it didn't show that option) I chose Windows 8.  It gives me further instructions as showing in the following picture:    Here is the problem, there is no sharing tab (see the following screen shot)   Method 2: As described in   I learned on this site that it is called Reverse tethering. I read and followed the procedure mentioned in    I followed all the steps described in that answer, until executing the command.   Install USB drivers from Android SDK [DONE] Connect USB cable and activate USB Tethering (You should see on a new network interface.) [DONE] Bridge the 2 network interfaces [DONE] On your computer execute adb shell netcfg usb0 dhcp [PROBLEM]   When I execute adb shell netcfg usb0 dhcp it says: error: device not found. I tried the same thing with USB Debugging in Developer settings, ON &amp; OFF. The error remains the same.  Method 3: Rooting the device and using an app  I read here and there that there is another method, in which we can directly use an app but it requires rooting the device. My phone is a new phone and warranty becomes void if I root it.   EDIT: I've tried the method suggested in answer, at the 4th step of method 2, I'm facing this problem: