Can anyone give me an example of one? I know it's equivalent to finding a noetherian that isn't PID, but I'm not sure!
As stated in the question, I am looking for a Noetherian ring over which the Euclidean algorithm is not valid.  I am trying to construct non-trivial examples of Noetherian rings.  Thank you.
Find the Limit (n to infinity) Hi I have a question regarding of limits to infinity please help! Thank You!  The question states the user to find the following limit:  $  \lim_{n\to\infty} n^2 ({\sqrt[n]{x}-\sqrt[n+1]{x}}) $      Where $(x &gt; 0) $  Thank You!!! I am designing gamecards in LaTeX  Everything is going pretty ok so far but when i try to insert the %-sign for a question about percentages my file keeps giving errors.  I tried putting the percentage between dollarsigns because i know just % will make it a 'remark' and that the dollarsign makes it 'mathematical data'.   Whats the reason for the error?   P.s. the error says: file ended while scanning use of \kaart. ...    Codesample:   {\kaart{Bekijk statistiek kaart F:   Schat het percentage mensen dat als lievelingseten spaghetti opgaven, je mag er slechts $ 2% $ langs zitten. } { $ 11% $}} While dabbling in Clojure I've written a very basic program to echo whatever the user types into it. However, it doesn't run in a way that I'm perceiving to be natural. Here's the code:  (defn goo []   (print "echo&gt; ")   (def resp (read-line))   (print resp) )   I would expect the code to run like this (for me typing in foo as the input to read-line):  user=&gt; (goo) echo&gt; foo foonil   But instead, the echo and read-line is switched:  user=&gt; (goo) foo echo&gt; foonil   Why does this happen? Is there a subtlety I'm missing?  EDIT: From Joe's answer, the updated correct solution is:  (defn goo []   (print "echo&gt; ")   (flush)   (def resp (read-line))   (print resp)   (flush) )   Also, the flushes aren't necessary if you use println instead of print. Seems that I just proved $2=4$. Adventurer enters an other-dimensional realm where he repeatedly dies unexpectedly and wakes up in reality $G \times H \cong G \times K$ , then $ K \cong H$ I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness of factorization of finitely generated abelian groups.  My questions are   If group $G$ is finite,$G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$? Can you give me a easiest proof and it can be proved by projection function on external direct product? If $G,H,K$ are groups.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. This statement is false. What is a counterexample?   *$\times$ is external direct product How did Ron speak Parseltongue and re-enter the Chamber of Secrets? Although 'The Deathly Hallows' was overall an excellent conclusion to Harry Potter's saga, there were moments within the book that felt infuriatingly lazy, like Rowling was whipping the plot along regardless of the holes she tore in it. For me, the lowest point was when Ron gargles at the entrance of the Chamber of Secrets- and succeeds!- so he can get some Basilisk fangs to destroy a Horcrux (which are conveniently vulnerable to this venom).  This is, by itself, a shockingly slapdash deus ex machina, but even worse is how it is completely inconsistent with the guidelines Rowling herself set. First and foremost, Parseltongue is not just a language, but a magical ability that can only be learned through hereditary or magical means (Potter being the sole example of the latter). This is firmly established by the fact that even Dumbledore, the greatest wizard of all time and a polyglot, is completely unable to speak it despite understanding it perfectly.  Without linguistic knowledge nor inherent ability, the chances of making any kind of coherent sentence (let alone the right one!) would be incredibly slim even in the case of mundane languages; it would be like gibbering at a Filipino in the hope that what you said was fluent Tagalog and didn't insult their mother.  Other baffling issues somewhat related to my question are thus; even if Ron had somehow unlocked the gate, how did he make his way down? In the second novel, Lockheart's foolery demolished the main tunnel to the Chamber, making it impossible to access or leave; the fact that the teeth are still there also implies that the Chamber wasn't excavated and the rubble cleared in the interval. Granted, he could have tried Apparating, but then he wouldn't have needed to fake Parseltongue to begin with. What are the origins of the word "nice"? How do I change the GRUB boot order? Easy way to change apache www location How do file permissions work? How does Salesforce SSO call just-in-time handler methods Is there a way to log all Microsoft SQL Server queries executed on a certain server, with their respective durations?  I'd like to be able to analyze all queries will a tool like what  does for PostgreSQL. While we're at it: do you know of any such tools? Why does my object not show up? This is meant to solve this problem once and for all. What is every reason an object isn't showing up in a render or viewport in when using the cycles render engine? how to remove the double quotes in a csv I have a CSV file like  a.csv "1,2,3,4,9" "1,2,3,6,24" "1,2,6,8,28" "1,2,4,6,30"   I want something like   b.csv 1,2,3,4,9 1,2,3,6,24 1,2,6,8,28 1,2,4,6,30   I tried awk '{split($0,a,"\""); But did not help.Any help is appreciated. When does the set enter set theory? Does Earth's Rotation Affect Its Shape? The question I am working on is, &quot;Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?&quot;  I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape? How to: Unlimited Bash/shell history? Is there a way to define an unlimited history in Bash ? Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in $U$? Mongodb: Possible to enlarge a capped collection? If a collection is created as capped, is it possible to increase the cap without losing data?  Or to increase the cap without recreating the whole collection?  Thanks