In the last Algebraic Number Theory lecture we were proving that a Noetherian domain $A$ is a DVR iff it is intergrally closed and it has a unique non-zero prime ideal. At some point the lecturer used an argument that if $A$ is Noetherian, then $\textrm{End}_A(\mathfrak{p})$ is Noetherian as an $A$-module (where $\mathfrak{p}$ is the unique non-zero prime ideal).   Why is what I described above true? (If it is.) Is a more general statement true: if we exchange $\mathfrak{p}$ with any Noetherian module $M$, will $\textrm{End}_A(M)$ be Noetherian as an $A$-module?
Let $A$ be a ring that is not a field and $M$, $N$ modules over $A$. What conditions on $M,N$ (and maybe also on $A$) guarantee that $\operatorname{Hom}_A(M,N)$ is finitely generated? Is it enough to assume $M,N$ finitely generated?
My question is, how would I make the visible HDRI/Enviroment Map be image 1 and then have reflections as Image 2? If someone can post a node setup, or a .py file that would be great &lt;3 I was thinking maybe using the Is Camera Ray node but I'm not confident it would work, and have no idea where to start. In GMAIL you can list all your boss-tell-off and wife-cuss-out emails by clicking Drafts    I'd like to see my started-but-never-finished responses, possibly between x Answers and y Votes in /&lt;&lt;user&gt;&gt;/Stats/? Related post   In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory.  Consider commutator $$ [ \phi(x), \phi(y) ] = D(x-y) - D(y-x) \tag{2.53} $$ where $D(x-y)$ is the two-point correlation function,  $$D(x-y):= \langle 0 | \phi(x) \phi(y) | 0 \rangle  = \int \frac{d^3 p}{ (2\pi)^3} \frac{1}{ 2E_{\mathbf{p}}} e^{-ip(x-y)}\tag{2.50}$$  P&amp;S argued that each term in the right-hand-side of (2.53) is Lorentz invariant, since $$\int \frac{d^3p }{ (2\pi)^3} \frac{1}{2E_{\mathbf{p}}} = \int \frac{ d^4 p }{ (2\pi)^4} (2\pi) \delta(p^2-m^2)|_{p^0&gt;0} \tag{2.40}$$ is Lorentz invariant.  Since there exists a continuous Lorentz transformation in the spacelike interval $(x-y)^2&lt;0 $ such that $(x-y) \rightarrow - (x-y) $ and $D(y-x)=D(x-y)$, (2.53) equals zero in the spacelike interval. In timelike interval, since such continuous Lorentz transformation does not exist, (2.53) is non-zero in general.  My question is, consider a non-continuous Lorentz transmation in the timelike interval, $PT$, namely time reversal times parity transformation. I can also let  $(x-y) \rightarrow - (x-y) $. Why (2.53) in the timelike interval is non-zero?  I guess $PT$ will let (2.40) go to $p^0&lt;0$ branch. But I am not sure if it breaks the Lorentz invariant of (2.40) and (2.50). Built-in text editor in Terminal Is there a built-in text editor in the terminal on Lion?  (Like, for example, nano on Linux.) Is it "5–6 weeks are a lot of time" or "5–6 weeks is a lot of time"? I was just copyediting  and my native English speaker Sprachgefühl told me I had to correct the grammar of one sentence:     ... 5–6 weeks are a lot of time ...   by changing the are to is. But as I was doing so I started wondering why is it that in this case it seems that I have to make the verb disagree with the plural subject?  So is my feeling for English going bad or if I did the right thing, how could I explain this to somebody who's learning English for instance? I'm seeing red dots everywhere On all Stack Exchange sites where I have review privileges, the review queue menu icon has a red dot. No matter what I try, it doesn't go away.     on Meta Stack Overflow might or might not be related. Bash auto_completion with Xubuntu and xrdp from windows I'm using Xubuntu and connecting over RemoteDesktop with Windows 7.  Everything works fine, but the auto_completion in my bash doesn't work. I checked the .bashrc file in my home directory and also the /etc/bash.bashrc.  I uncommend the lines below in my local bashrc configuration and also in bash.bashrc, but nothing changes.  if ! shopt -oq posix; then   if [ -f /usr/share/bash-completion/bash_completion ]; then     . /usr/share/bash-completion/bash_completion   elif [ -f /etc/bash_completion ]; then     . /etc/bash_completion   fi fi Customize Ubuntu for a library Internet kiosk Why do hard links exist? I'm trying to wrap my head around how SQL Server decides to grant certain rights based on a users login privileges.  If a user is part of a group "READ ONLY" which has access to DB1, but they are also a part of a group "SysAdmin" which has full db owner rights to the same DB1, which rights are they given when they log in?  On a similar note. What if they are part of a group but also have a standalone account with other middle of the road privileges? How to create multiplied identity t-sql? Nonzero nilpotent elements in Zn So I've been stuck on this proof for the past threes days now. I keep going back and forth and making no progress.     Let $n\in \mathbb{Z}_{\geq 2}$. Show that $\mathbb{Z}/n\mathbb{Z}$ has a nonzero nilpotent element iff there is a prime $p$ s.t. $p^2$ divides $n$.   $(\Rightarrow)$ I know $a$ can be factored into distinct primes, so $a=p_1^{e_1}\cdot ...\cdot p_m^{e_m}$ then $a^k=p_1^{e_1k}\cdot ...\cdot p_m^{e_mk}$.  Also since $n|a^k$ so $a^k=nd$ for some $d\in \mathbb{Z}$.  $a^k=p_1^{e_1k}\cdot ...\cdot p_m^{e_mk}=nd$. So $nd$ has that distinct prime factorization.  Then I get lost on what to do. Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) What hardware does Ubuntu Touch support? Reading list in topological QFT Modified files in a git branch are spilling over into another branch I am working on a git repository with a master branch and another topic branch. I have switched to topic branch and modified a file. Now, if I switched to master branch that same file is shown as modified.   For example:  git status in git-build branch:  # On branch git-build # Changes to be committed: #   (use "git reset HEAD &lt;file&gt;..." to unstage) # #       modified:   cvsup_current #   Switch to master branch  [root@redbull builder_scripts (git-build)]# git co master M       builder_scripts/cvsup_current Switched to branch "master"   git status in master branch  [root@redbull builder_scripts (master)]# git status # On branch master # Changes to be committed: #   (use "git reset HEAD &lt;file&gt;..." to unstage) # #       modified:   cvsup_current #   Why is that the file is shown as modified in master branch even though it was modified in git-build branch?   My understanding was that the branches are independent of each other and when I change from one branch to another the changes do not "spill over" from one branch to another. So I am obviously missing some thing here.   Anyone got a clue stick? Join matrix based on conditions How to remove a repository? $ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$ How can I prove that $\,ax\equiv b\pmod{\! n}\,$ has a unique solution if $\,\gcd(a,n) = 1$?  I suspect that the Bezout identity for $\,\gcd(a,b) = 1$ is the key but I don't see how to proceed. Image upload from photo gallery So this is pretty obvious. Image upload?    How about allowing to attach images in questions from the photo gallery, by perhaps sending the image for off-site storage like done from the full-web version?