Given $R$ an integral domain with field of fractions $F$, I want to prove $F$ is a finitely generated $R$-module if and only if $R = F$.   For $(\Rightarrow)$: I tried to first define a generating set based on the definition i.e. finitely many $m_1,\dots,m_n$ such that $\sum_i Rm_i  = M$. Then somehow show every element in $R$ is invertible. From where $D=F$ but how...  For $(\Leftarrow)$: If $R = F$, I am not sure, but I know by definition, the embedding of $R$ in $F$ maps each $n \in R$ to the fraction $\frac{an}{a}$ for nonzero $a$ in $R$. In further details, I am a bit lost though.
I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that $1\not\in D$.)    My original plan was to localize at a maximal ideal and then use Nakayama's lemma but as we don't have a $1$ our localization may not be local (in fact we need not have maximal ideals).  So I'm a bit lost as to where to go?  Thanks for any help.
Mod of numbers with large exponents [modular order reduction] I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great. Can you take the same feat multiple times? How to store date/time and timestamps in UTC time zone with JPA and Hibernate TL:DR Smart guides aren't accurate - even though the snap, (end up causing more problems in the long run) is there a way to fix it?  Maybe there are some settings that I'm missing (need to be changed)    I use my smart guides all the time (and am overall pleased with them). I do have my issues with it, as explained here:  A lot of times, when I'm trying to make a bunch of triangles like this:    Done by making a triangle, cloning, flipping, rotating etc. Repeat until full  (Perfect) Snapping using Smart Guides    Ends up looking likebetween the triangles (Outline Mode)    Now I personally (with a strong attention to detail) will notice that there is something off after making just 1 or 2 triangles (Even before outline mode... strokes look oh so slightly thicker/off)   And then I check in outline mode... and sure enough:    Notice the line spaces (and this is in outline mode, so if I didn't notice it before, now it's sitting right in front of my nose)  The way I (seemingly) "fixed" the first image was by adding in a thick stroke outline so that masks out the difference, however, in a lot of my cases I don't want to have a stroke at all (and then you really notice the problems). Registered Apple Developer Agreement when under 18 How is MSE divided by $\sigma^2$ always a chi-square distribution? Suppose we have a situation like      A ball of some mass $m$ with some velocity collides with rod hinged at point $A$. Is momentum conserved in this situation? I know that hinge will give impulsive force on rod but that is internal force when we take ball and rod together as a system so that shouldn't stop us using conservation of momentum equation How do I vertically center a single cell within a table? What are all possible ways to stop a zombie from despawning? Is there any way for which we are able to stop a zombie from de-spawning no matter how far we go? If so, what are all possible ways?  Bonus question: what are all the other techniques for which we can prevent a zombie from de-spawning within a specific block radius? I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.  In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?  Thanks! What is the emergency phone number in different countries? Placement of the asterisk in pointer declarations I've recently decided that I just have to finally learn C/C++, and there is one thing I do not really understand about pointers or more precisely, their definition.  How about these examples:   int* test; int *test; int * test; int* test,test2; int *test,test2; int * test,test2;   Now, to my understanding, the first three cases are all doing the same: Test is not an int, but a pointer to one.  The second set of examples is a bit more tricky. In case 4, both test and test2 will be pointers to an int, whereas in case 5, only test is a pointer, whereas test2 is a "real" int. What about case 6? Same as case 5? A field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conservative? Can someone please give me an intuitive explanation and insight into this, and if it is true, why? Also, please try to not be too rigorous (only in grade 9). Let $({\Bbb R},{\mathcal A},\mu)$ be the measure space where ${\mu}$ is the Lebesgue measure.  Assume that $\int_{\Bbb R}|f|^pd\mu&lt;\infty$ ($p\geq1$). There is an exercise for proving that       $$\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s).$$   If one formally let $s=t^p$, then $ds=pt^{p-1}dt$ and one would immediately get the equality above. However, how can I justify it rigorously in the sense of Lebesgue integration?    [Edited] What puzzles me is that I have the "change of variable" theorem for Riemann integration, but all the integrations here are in the Lebesgue sense. Unless one has shown that the integrands are also Riemann integrable, how can I directly use "change of variable"? How to check if installing a package will require a reboot? $A$ is a $n\times n$ matrix such that $ A^m = 0 $ for some positive integer $m$.   Show that $A^n = 0$.   My attempt: For $n &gt; m$, it's obvious since matrix multiplication is associative.  For $n &lt; m$, $A^n\times A^{m-n} = 0$; not sure what to do next. Also I know that $\det A = 0$. Create Random Number Sequence with No Repeats Duplicate:    I want an pseudo random number generator that can generate numbers with no repeats in a random order. For example: random(10) might return 5, 9, 1, 4, 2, 8, 3, 7, 6, 10 Is there a better way to do it other than making the range of numbers and shuffling them about, or checking the generated list for repeats?  Edit: Also I want it to be efficient in generating big numbers without the entire range.  Edit: I see everyone suggesting shuffle algorithms. But if I want to generate large random number (1024 byte+) then that method would take alot more memory than if I just used a regular RNG and inserted into a Set until it was a specified length, right? Is there no better mathematical algorithm for this. How does "this" keyword work within a function? How can I list the contents of a .zip folder in c#? How can I list the contents of a zipped folder in C#? For example how to know how many items are contained within a zipped folder, and what is their name? Get total duration of video files in a directory I have a list of .ts files:   out1.ts ... out749.ts   out8159.ts  out8818.ts   How can I get the total duration (running time) of all these files?