Let $A$ be a finitely generated $k$-algebra with no nilpotents. What do I need to show in order to prove it's a finite dimensional vector space over $k$?   For example, is it enough to show that $A$ has only finitely prime ideals and every prime ideal is maximal?
Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:   (1) $A$ is Artinian.   (2) $A$ is a finite $k$-algebra.   I have a question in the proof of (1$\Rightarrow$2): By using the structure theorem, we may assume that $(A,m)$ is an Artin local ring. Then $A/m$ is a finite algebraic extension of $k$ by . Since $A$ is Artinian, $m$ is the nilradical of $A$ and thus $m^n=0$ for some $n$. Thus we have a chain $A \supseteq m \supseteq m^2 \supseteq \cdots \supseteq m^n=0$. Since $A$ is Noetherian, $m$ is finitely generated and hence each $m^i/m^{i+1}$ is a finite dimensional $A/m$-vector space, hence a finite dimensional $k$-vector space.  But now how can I deduce that $A$ is a finite dimensional $k$-vector space?
How to forward specific log file outside of /var/log with rsyslog to remote server? How can I forward message from a specific log file like /www/myapp/log/test.log with rsyslog client to remote rsyslog server? This log file is outside of the directory /var/log. I have seen many folks doing this  concluding that shadow from point A on the Moon can travel to point B on the Moon faster than light.  What I fail to understand here (and I am sure I am wrong), is nothing can travel faster than light. I also read about Relativity where Einstein stated that every event happening for you depends on how fast the light gets from where the event is happening to you.  This is my particular problem with this Moon shadow experiment. You cannot make a shadow travel faster than light, because in order for the "disappearance" of light to get to the Moon, it has to travel at light speed.  Here is one way to understand my point. Consider we shine a flashlight on Moon, like the guy in the linked video did, we move the finger across the face of the flashlight, before we moved the finger it was casting the shadow at Point A on Moon, after we finish moving our finger, the shadow is at Point B on the Moon.  When the finger reaches the end of it's moving length, the light waves that have left the flashlight before the finger reached there will have to hit Point B and it will take them a second and a half to get there, and another second and half for us to see that shadow on point B, which makes it 3 seconds + finger moving time for us to actually see the shadow at point B. That means shadow traveled (say) across the diameter of the Moon in 3 seconds.   So. about 3000 Km in ~3 seconds make it 1000 Km per seconds which is very tiny compared to the speed of light. How can these experiments conclude that Shadow could be made to travel faster than light? What is the obvious clue that I am missing here? I have critical low storage level and can't understand what takes much of it and how that's possible that the installed applications can use 7 GB? I only use a few social media clients without any games or soft like Photoshop. Also on the second screenshot it indicates that the apps are using just 1.13 GB and so I can't understand the situation. Also what is the "Other"? Local explanation of the Aharonov-Bohm effect in terms of force fields Here is an interesting paper for the Physics SE community:      On the role of potentials in the Aharonov-Bohm effect. Lev Vaidman. .  [quant-ph].   You should check it out because it's an amusing read, but I will summarise the argument to whet your appetite, assuming you have basic familiarity with the . In the traditional AB setup, one considers an electron in a superposition of paths, taking it in two opposite directions around a solenoid treated as a classical source of the electromagnetic field. The observable relative phase acquired between the electron's paths is attributed to the influence of the magnetic vector potential on the electron, which cannot be globally gauged away - despite the absence of a physical field anywhere along the electron's path(s) - due to a topological obstruction.  Instead, Vaidman considers the effect of the electron on the sources of the field, treating the latter as quantum particles. He shows that the relative phase between the two branches of the wavefunction can be considered as arising from the action of the physical field of the electron, which is not zero at the position of the sources. However, Vaidman uses highly contrived gedankenexperiments and completely semi-classical arguments, which begs a pair of concrete and related questions.  1) Can Vaidman's first, electric AB effect gedankenexperiment be described in a fully quantum manner, by solving (at least approximately) the three-particle Schroedinger equation? If not, why not?  2) Is it possible to explain within this formalism the experiments of Tonomura et al. (), who used a superconductor to completely shield the magnetic field of the source? WIth Vimeo Plus account, how to hide/remove the "play" button in the middle of an embedded video? Let $X_1$ and $X_2$ be two independent exponential random variables with the PDFs $f_{X_1}(x_1)=\lambda_1 \exp(-\lambda_1 x_1)$ and $f_{X_2}(x_2)=\lambda_2 \exp(-\lambda_2 x_2)$.   Let $X= \frac{X_1X_2}{X_1+X_2+a}$ be a r.v. for which I want to derive the CDF, where $a$ is a positive constant.  In other words, I need to calculate $\mathbb{P}\{X &lt;x \}$.  Any ideas or hints?  Edit (attempt):  Using a similar approach to that used , we have $X_1 \in [0, \infty)$ and  $X_2 \in [0, \frac{x(x_1+a)}{x_1-x})$. The latter interval results from: $\frac{x_1x_2}{x_1+x_2+a} &lt; x$, thus $x_2(x_1-x)&lt;x(x_1+a)$ which implies $x_2 &lt; \frac{x(x_1+a)}{x_1-x}$; but I am not sure if this is correct since $x_1-x$ can sometimes be negative (?). Based on the above, we can write: \begin{align}  \mathbb{P} \{ \frac{X_1X_2}{X_1+X_2+a} &lt; x \}  &amp; = \int_{x_1=0}^\infty \int_{x_2=0}^{ \frac{x(x_1+a)}{x_1-x}} \lambda_1 \exp(-\lambda_1 x_1) \lambda_2 \exp(-\lambda_2 x_2)  dx_2 dx_1  \\ &amp;= \int_{x_1=0}^\infty \left(1-\exp(-\lambda_2\frac{x(x_1+a)}{x_1-x}) \right) \lambda_1 \exp(-\lambda_1 x_1) dx_1  \\ &amp; = 1-\lambda_1 \int_{x_1=x}^\infty \exp(-\lambda_2\frac{x(x_1+a)}{x_1-x} ) \exp(-\lambda_1 x_1)   dx_1 \end{align} In the last equality: for the integral of $\lambda_1 \exp(-\lambda_1 x_1) dx_1$, I suppose that $x_1 \in [0,\infty)$, whereas for the second term I suppose that $x_1 \ge x$ (since $x_2$ should be positive) (?).  Is my approach correct ? Array of zero length I am working on refactoring some old code and have found few structs containing zero length arrays (below). Warnings depressed by pragma, of course, but I've failed to create by "new" structures containing such structures (error 2233). Array 'byData' used as pointer, but why not to use pointer instead? or array of length 1? And of course, no comments were added to make me enjoy the process...  Any causes to use such thing? Any advice in refactoring those?  struct someData {    int nData;    BYTE byData[0]; }   NB It's C++, Windows XP, VS 2003 What is the "exposure triangle"? Given this XML, what XPath returns all elements whose prop attribute contains Foo (the first three nodes):  &lt;bla&gt;  &lt;a prop="Foo1"/&gt;  &lt;a prop="Foo2"/&gt;  &lt;a prop="3Foo"/&gt;  &lt;a prop="Bar"/&gt; &lt;/bla&gt; The draft feature was discussed a lot  and in other posts, but I didn't see much talk about the OP's mention of being able to have multiple.     Optionally, allow multiple draft questions and a page to edit/delete them.    I think the draft feature where it autosaves and stores it server-side is great, but in most things where they have a draft feature (like email clients), they allow you to have multiple drafts (also stored server-side).  The expiration can even work the same as it does for the single draft system right now.  You wouldn't even need full page to store them, I think the user profile page could have a tab for it like it currently has for answers, questions, and everything else.    It could look very similar to how questions look, with it's title and tags listed there. What is the use of percent signs (%) at the end of lines? (Why is my macro creating extra space?) I see that the code in many packages and examples contains percent signs % at the end of (many) lines. What are they used for? Do they affect the parsing of those lines? I submitted a new version of the sitemap index for my site in Google Search Console two days ago. Soon after submission, GSC started saying that the index was "processed successfully", and yet has continued to display zero as the number of discovered URLs.  Note that my sitemap index points to about 11000 sitemaps, which provide actual links to pages.  I'm pretty sure the format of both the sitemap index and the individual sitemaps is correct.  Screenshot:    Should I worry about this? What does it mean?  Edit on 2020-02-03:  Google finally started displaying the right number in Total discovered URLs. It must have happened sometime in January. Why so late - I have no idea... Should I be offered the option to upgrade to 18.04.1? I tried Libreoffice, but its Import option while pasting in the cell is very  annoying.I also tried Gnumeric but I am not able to paste using middle button.Parcellite to make copy paste faster and middle button pasting is not working with Gnumeric. Is there any simple table maker app available that can accept only simple unformatted text? All these software are very complicated for this simple work. Creating mask file from shapefile using ArcGIS for Desktop? I have a shapefile of Lakes over my study area. Each polygon has attribute information of each lake, name, shape etc. Actually I want to eliminate all lakes from analysis. In other word, I want lakes to have 0 value and the area that out of boundary of lake to have their orginal pixel values. when I do subset using this shapefile, the output is not I want because my shapefile deoesn't have any value outer boundary of lake.   How can I build a proper mask file (inner part of lake has 0 value, outer is 1) using this shapefile in ArcGIS Desktop? I started to read about Diffie-Hellman exchange and I encountered a problem which I don't know how to solve.  Given $g=3$, $p=131$, $g^x\bmod p=112$, $g^y\bmod p=74$, compute $g^{xy}\bmod p$   $113$ $110$ $112$ $111$   How could I compute $g^{xy} \bmod p$? Or at least verify which one is correct. Any idea? Visas to travel the world I understand that you have to have visas to enter some countries, but I was wondering whether you have to pay for each visa separately or you can get a visa to go everywhere. You see videos and lists on what you should pack to travel the world, but no lists on the documentation needed. Create taxonomy terms on the fly Are Mersenne prime exponents always odd? I have been researching  so I can write a program that finds them. A Mersenne prime looks like $2^n-1$. When calculating them, I have noticed that the $n$ value always appears to be odd. Is there confirmation or proof of this being true? Proof of an inequality by induction: $(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + x_n$