Let $A \subset B$ be commutative integral domains with $\operatorname{Quot}(A) =  \operatorname{Quot}(B).$  Now consider the multiplicatively closed subset $S = \{x\in A\setminus\{0\}: x^{-1}\in B\}$.   I want to show that $S^{-1}A = B$. I would appreciate any help.
Let $R$ be an integral domain and let $S$ be a ring with $R \le S \le \text{Frac}(R)$ (fraction field).   Question: Is there a multiplicatively closed subset $U \subseteq R\setminus \{0\}$ such that $S=R[U^{-1}]$ ?
Earth Engine convert list with coordinates and values into a feature collection for export I need to create a feature collection from a list containing coordinates and band values.  These are extracted from a landsat image as shown in this linked script.    I'm unable to find a solution in creating a feature collection.  Here is the tail end code.  var valuesList = joinedImage.reduceRegion({   reducer: ee.Reducer.toList(4),   geometry: myGeometry }).values().get(0);  var feature = ee.Feature(null, valuesList);   // Wrap the Feature in a FeatureCollection for export. var myFeatures = ee.FeatureCollection([feature]);   // Export the image, specifying scale and region. Export.table.toDrive(myFeatures, "data", "myData", "B4-B5",  "CSV");   This doesn't work, because it is clearly the wrong way to create the feature collection.  I've tried to create it similar to Rodrigo E. Principe's solution of mapping a Feature Collection but I can't seem to create the initial feature collection to start with. Word for a person who continually tries to teach you when it's not necessary I've experienced this way too many times, and the best example I can give is a person who treats you like you have no idea how to do X and who keeps teaching you how to do it. I feel handicapped. Take for example how to play a card game. You know how to play the game because you've played it 5 times. However, there is a person always insists on explaining the game to you - like you are always new to the game - when you don't actually need any help and you just want to think for yourself. Another example situation is doing a repetitive task which you already have experience with, and yet the person continually tries to 'teach' you when you don't need help/already know how to do it. It almost makes you mad because you really don't need help. As an example which recently happened to me:  &quot;Hey, do you remember how to start the wood chipper?&quot; &quot;Yes I do&quot; &quot;Okay, well, let me show you one more time&quot;  Again, is there a term for someone who does this? Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$ Prerequisites for Algebraic Geometry Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch?  If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic geometry with my state of knowledge?  Thanks It is me who am to blame.    It is I who am to blame.      Which one is correct here? I am bewildered about these two sentences. Let $\mathbb{S} = \{ (X,Y,Z) \in \mathbb{R}^3 :X^2 + Y^2+Z^2=1\}$ denote the Riemann sphere. The following problem is taken from Gamelin "Complex Analysis" chapter one:     What is the image of the spherical cap $ C := \{(X,Y,Z) \in \mathbb{S} : A \leq X \leq 1\}$, where $A$ is some fixed real number between $-1$ and $1$, under the stereographic projection $ \pi(X,Y,Z) = \frac{X+iY}{1-Z}$?   Here is my work so far:  ( The case $A=1$ is trivial, so assume $A &lt;1$)  We want to find $\pi(C) = \{z \in \mathbb{C} \mid \exists(X,Y,Z) \in C: \pi(X,Y,Z) = z \}= \{z \in \mathbb{C} \mid \pi^{-1}(z) \in C\}$, where the last equality holds by $\pi$ being a bijection. Write $z=x+iy$.  Here, $ \pi^{-1}(z) \in C\iff A\leq \frac{2x}{1+x^2+y^2} \leq 1 \iff (\star) :A \leq \frac{2x}{1+x^2+y^2}$, since the inequality $2x \leq 1+x^2+y^2$ holds for all $x,y \in \mathbb{R}$.  $A=0$: $( \star)$ reduces to $ 0 \leq x$, so in this case the image is $\{z \in \mathbb{C} \mid \Re(z) \geq 0 \}$.   $A &gt;0$: Since $A \neq 0$ in $(\star)$ we have,  $ \frac{2x}{1+x^2+y^2} \geq A \iff Ax^2-2x+Ay^2+A \leq 0 \iff (x-\frac{1}{A})^2 +y^2 \leq \frac{1}{A^2}-1$. Thus the image is the interior and boundary of the circle centered at $\frac{1}{A}$ with radius $\sqrt{\frac{1}{A^2}-1}$. Note $\frac{1}{A^2}-1 \geq 0$ as $A \in (0,1)$.  $A &lt;0$ : By symmetry and the above, here the image is $ \mathbb{C} / \{ $ interior and boundary of circle centered at  $  \frac{1}{A}$ with radius $\sqrt {\frac{1}{A^2}-1} \}$.  Is my working correct? Many Thanks! I have the age-old problem of a div wrapping a two-column layout. My sidebar is floated, so my container div fails to wrap the content and sidebar.  &lt;div id="container"&gt;   &lt;div id="content"&gt;&lt;/div&gt;   &lt;div id="sidebar"&gt;&lt;/div&gt; &lt;/div&gt;   There seem to be numerous methods of fixing the clear bug in Firefox:   &lt;br clear="all"/&gt; overflow:auto overflow:hidden   In my situation, the only one that seems to work correctly is the &lt;br clear="all"/&gt; solution, which is a little bit scruffy. overflow:auto gives me nasty scrollbars, and overflow:hidden must surely have side effects. Also, IE7 apparently shouldn't suffer from this problem due to its incorrect behaviour, but in my situation it's suffering the same as Firefox.  Which method currently available to us is the most robust? I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.   $\dfrac {x}{0}$ is Impossible (  If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only $2. $ has infinite solutions so as $1.$ has none solution, how and why ?) $\dfrac {0}{0}$ is Undefined and has infinite solutions. (How come one be Undefined and yet has infinite solutions ?) $\dfrac {0}{x}$ and $x \ne 0$, it's okay for me, no problem, but if someone else wants to add something about it, feel free to do it. Why is the following C# code not allowed:  public abstract class BaseClass {     public abstract int Bar { get;} }  public class ConcreteClass : BaseClass {     public override int Bar     {         get { return 0; }         set {}     } }      CS0546 'ConcreteClass.Bar.set': cannot override because 'BaseClass.Bar' does not have an overridable set accessor We will make the convention that exact categories be skeletally small.   Is this construction (used in a mathematical context) correct? There is something that strikes me as odd in that "be". Should it be "are"? I know I've seen a similar way of using "be" like that, without conjugating the verb, but I'm not completely sure when it should be used. Stopping to eat is tedious, is there any way to automate the process? While I've managed to optimize and automate almost everything, constantly having to stop and eat is annoying (I run around a lot, so I get hungry quickly). Is there any device or tool that will keep me well fed at all times without having to stop and feed? How to split a very long equation (using AMSMATH package)? Are moving objects producing stronger gravity fields? If the strength of gravitational influence exerted by a body is derived from its mass and energy then is it true that a moving object which has some kinetic energy should also produce stronger gravitational pull than a static/slower one?  If that's correct then would it be also possible for an object moving fast enough to produce gravity so strong that it would collapse into a (fast moving) black hole? I'd like to know if there is a way to run  program/shell script without typing full path:  /opt/idea/bin/idea.sh How do you deserialize json properties that are reserved words in apex? Incorrect whitespaces or list item size in Android SE app Noticed that sometimes when there are 10+ answers to a question - some answers look like they have \n\n...\n line breaks in the end, so the body of the answer has incorrect height.  It probably has something to do with ViewHolder or adapter not setting correct row height. Although as far as I know, it should do this according to text amount/length. I recently received an old iMac secondhand.  I have no way to reach the original owner. I am trying to upgrade the OS software, but the prompts demand the administrator's name (which is auto-filled) and password (which I do not have).  I understand why these security features are in place, but is there any way in the world for me to access Admin. privileges without the password? Mankind prepares for a war with aliens by warring among itself I am looking for the title or author of a atory I read back in the 90's, no idea when it was written. The plot goes something like this: Humanity has gained access to space travel and sends 9 or 12 survey ships towards galactic center, only 1 returns with stories of an enemy so rapacious that humanity starts preparing for war. After a long time, the enemy hasn't shown up and people begin to doubt the need to keep building for war. The government engineers a plan to turn the core systems against the fringe worlds and start warring amongst ourselves so that we may be prepared for when the real enemy shows up.  Written with the focus on one spaceship captain battling in our own internal war. Permissions of /Users keep changing back to drwxrwxrwx To put it simple, there's a simple java swing app that consists of JFrame with some components in it. One of the components is a JPanel that is meant to be replaced by another JPanel on user action.  So, what's the correct way of doing such a thing? I've tried  panel = new CustomJPanelWithComponentsOnIt(); parentFrameJPanelBelongsTo.pack();   but this won't work. What would you suggest?