I have this exercise:     Let $R$ be a Noetherian ring. Let $S\subseteq R$ be a multiplicatively closed set. Prove that $S^{-1}R$ is a Noetherian ring.   First off we claim that given a subset $I\subset R$, $$S^{-1}I \lhd S^{-1}R \quad \Rightarrow \quad I\lhd R\ \land \ I\ \cap S=\emptyset $$ That's true because $$ S^{-1}I \lhd S^{-1}R \quad \Rightarrow \quad (\ \frac {i_k} s \in S^{-1}I \Rightarrow \exists \ d_k\ \mathrm {maximal} :d_k|i_k,d_k\notin S)$$ So if we take the ideal in $S^{-1}R\ $generated by the elements $(\frac {d_1} 1,\frac {d_2} 1,\dots ,\frac {d_k} 1,\dots)$, clearly we obtain $S^{-1}I$, plus $I=(d_1,d_2,\dots,d_k,\dots)\lhd R$. Since every ideal on $R$ is finitely generated, it follows that ideals are finitely generated on $S^{-1}R$ too.  Is this correct? Do I need to be more exhaustive or am I wrong with something? Thank you :)
This is an unproven proposition I've come across in multiple places.   Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian.  Why is this? I thought about taking some chain of submodules $$ S^{-1}M_1\subset S^{-1}M_2\subset\cdots $$ and pulling back to a chain  $$ M_1\subset M_2\subset\cdots $$ of submodules of $A$ which must eventually stablize. Is there more to it than this? I kind of wary of assuming all submodules of $S^{-1}A$ have form $S^{-1}M$ for $M\leq A$.
Is there a safe way to run a diff on two zip compressed files? Seems this would not be a deterministic thing, or is there a way to do this reliably? Why is $\sqrt{-x}*\sqrt{-x}=-x?$ I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold.    This is exercise 2.11 from Atiyah and MacDonald's Introduction to Commutative Algebra, but unfortunately all the solutions available online are either very sparse with regard to this question or seem to use (to my surprise) a generalized version of Cramer's rule. Is there perhaps some other "cleaner" approach to solving this problem? My iPhone 5 won't boot anymore and when I plug it in my MacBook, it goes to recovery mode (cable logo => iTunes logo).  So I tried to restore it using iTunes but I keep getting this error    Any idea how to fix it?  While it was plugged in I tried holding power + home, it reboots the device and goes back to recovery mode and then any restore attempt results in a 2001 error as well. When it's not plugged in nothing will make it boot. Counting buffer overlap in QGIS For my thesis I'm modeling the identification of recreation areas.  Therefore, there has been a survey where people were asked to specify where they recreate. They had to indicate this place on a map and give a radius. Now I have to analyse which areas were most visited, and afterwards use the data to give a weighted rating with the travel distance.  So, I use QGIS to Buffer the data points by using the given radius. Now, I want to calculate for every point in the country how many times it was visited.  Any thoughts how to do this in QGIS? With the Superman lore, we know that light from a yellow star gives him his powers, light from blue stars makes him stronger, and light from red stars weakens him.  However, what happens if he is near a white dwarf, or a neutron star?  Has there been any evidence of Superman visiting a star system that wasn't red/yellow/blue? I failed to open a lock. Now what? UIRefreshControl without UITableViewController With powershell get exactly the same application list as in Add/Remove programms There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. Some of the ways I tried are:   WMI via Win32_Product and Win32Reg_AddRemovePrograms Registry, both x32 and x64 branches   For instance, I have 217 entries in AR and   gwmi  Win32_Product   returns 261 items.  gp HKLM:\Software\Microsoft\Windows\CurrentVersion\Uninstall\*, HKLM:\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\* | ? {![string]::IsNullOrWhiteSpace($_.DisplayName) } | select DisplayName   returns 119 items.   (gwmi Win32Reg_AddRemovePrograms) + (gwmi Win32Reg_AddRemovePrograms64) | sort DisplayName -Unique    returns 107 items.  I tried to nail it down with more complex filter that included InstallLocation, InstallSource, InstallDate etc without success. Do's and Don't's of Undergraduate Research I was fortunate enough to get a position as a researcher for the Mayo Clinic's SURF Program this year. My PI's lab focus is on the the immune system's role in CNS axonal and neuronal injury, specifically through the lens of how innate and adaptive immune effectors interact w/ infected neurons.  Although I do research under a professor at my college and I volunteered for a state university lab during the previous summer, this is my first REU/SURF opportunity, and I REALLY want to make a good impression. Here are my questions:  What are the do's and don't's in terms of being a skilled and efficient researcher?  Since I am still an undergrad, I know that I will be a less useful asset to the lab than a grad student or post-doc, but what can I do as an undergrad to not burden my colleagues and PI?  Thank you all for your help! Wishing you all the very best! Summary table with a field that shows components of each summarized record I have a table with 2 fields the first is for community and the second for the main community that it belongs to, as in the image  I want to summarize the second field (Belongs_to) but with an extra field (multiple value field) that shows components of each main community, as in the image  Could anyone help me please? Detach (move) subdirectory into separate Git repository I have a  repository which contains a number of subdirectories. Now I have found that one of the subdirectories is unrelated to the other and should be detached to a separate repository.  How can I do this while keeping the history of the files within the subdirectory?  I guess I could make a clone and remove the unwanted parts of each clone, but I suppose this would give me the complete tree when checking out an older revision etc. This might be acceptable, but I would prefer to be able to pretend that the two repositories doesn't have a shared history.  Just to make it clear, I have the following structure:  XYZ/     .git/     XY1/     ABC/     XY2/   But I would like this instead:  XYZ/     .git/     XY1/     XY2/ ABC/     .git/     ABC/ Show that any bounded linear functional on a normed linear space is continuous Show that any bounded linear functional on a normed linear space is continuous.   Can we say that it is uniformly continuous ?   Also, is it true, if we reverse the statement    any continuous  linear functional on a normed linear space is bounded ?? Rotate a point on a circle with known radius and position Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the circle so we will obtain a new point on the circle $C(x_c, y_c)$, like in the image below? How to calculate the $C$ coordinates?   Here the rotation angle is $90 ^ {\circ}$. In this example, $x_b = x_a$, $y_b = R$, $\alpha = 90 ^ \circ$. From the image we see that $x_c = R$ and $y_c = y_a$.  However, I want a general solution for any $A, B, R$ and $\alpha$. How can I use pathauto to generate URLs for deeply nested vocabularies? I am generating breadcrumbs from URL arguments as described .  I have a vocabulary that looks like this:  Arts-> Performing-> Dancing-> Ballet  I want my url to look like this: arts/performing/dancing/ballet  Using the pathauto module, I can do this for taxonomy terms:  [term:parent]/[term:name]   Which, on the Ballet page, gives me the following; not what I want:  dancing/ballet    If I use parents instead of parent,  ([term:parents]/[term:name]); this is close but not quite:  arts-performing-dancing/ballet   How can I get from arts-performing-dancing to arts/performing/dancing? Prove $\frac{1}{2}\sum_p \frac{1}{p^2}+\frac{1}{3}\sum_p \frac{1}{p^3}+\cdots$ converges Story identification - "bless the beasts that bless us" I'm trying to trace what was probably a short story rather than a full novel from my childhood.  I think it starts in the present day (so the 80s or thereabouts) and there was some kind of alternate reality bleed which meant there were (i think) ghost boars or pigs, they were becoming more physical with time, and so problematic. The family escaped when the family dog led them to and through a portal which they couldn't perceive to a safer place.  In the end, they ended up in some kind of rural community who had a religion / custom that required them to say "Bless the beasts that bless us" but some folk shortened it to BTBTBU which irritated the girl who was the principle character.  I used to read a lot of kids short story sci-fi and fantasy including Nicolas Fisk but I don't know whether this was from a compilation of several authors or a single author's collection. I installed Ubuntu 14.04.1 on one of my laptops and now I want distribute that exact installation to a few of my friends. I tried making an iso with Gparted on a live session but it makes a 500 GB file (I've only used around 8 GB) .   Thanks in Advance ! Prove that $2^{1/2}$ is irrational I'm trying to understand each part of this completed proof that my professor did, here is my interpretation in parentheses, please advise as necessary.  Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$  $$(p/q)^2 = 2 = p^2 / q^2 $$ $$\text{ (What was the purpose of squaring √2 and p/q?)}$$  $$p^2 = 2q^2$$ $$ \text{ (Is this basically just canceling out the q^2 on left side, and multiplying the right side?)}$$  2 | p^2 => (2 divides p^2 since it is a factor)  2 | p => (thus 2 divides p)  p = 2k for some integer k => (definition of even)  (2k)^2 = 4k^2 = 2q^2 => (Why are we squaring 2k, and how does 4k^2 = 2q^2?)   q^2 = 2k^2 => (Canceled out 2 on one side, and dividing on other)  2 | q^2 => (2 divides q^2)  2 | q => (thus 2 divides q)  2 | gcd(p,q) => (2 divides gcd of p and q)  2 | 1 Which is a Contradiction (no clue how I got here) How to keep the console window open in Visual C++? I'm starting out in Visual C++ and I'd like to know how to keep the console window.  For instance this would be a typical "hello world" application:   int _tmain(int argc, _TCHAR* argv[]) {     cout &lt;&lt; "Hello World";     return 0; }   What's the line I'm missing?