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Let $R$ be a commutative ring with identity, and $S$ a multiplicative subset of $R$ not containing $0$. Which step in my reasoning is wrong?   Every element of $S$ is a unit in $S^{-1}R$. Every element of $S$ that's a zero divisor in $R$ is a zero divisor in $S^{-1}R$. But zero divisors can't be units. Therefore it's not possible to localize $R$ at $S$. This appears to be related to the behavior originally suggested in .  It appears to be the same issue raised in , but this post has specific steps to reproduce the issue.  Here's what happened on one site in the network earlier today:   A question was posted much earlier Someone came along, and as an unregistered, new user with no other activity, posted a spam answer to that question at 04:36:54 UTC That same user posted a comment on their own answer with the same spam link at 04:37:31 UTC The answer was deleted as spam via community votes at 07:32:07 UTC A high-rep user saw the comment and raised a custom flag on it as being spammy at 07:32:54 UTC The unregistered user account was destroyed by a moderator at 08:13:10 UTC The comment now shows as "deleted by Anonymous" with a deletion time of 08:13:10 UTC The flag on the comment remains, but the only options are "Edit" and "Decline"; there is no "Delete" action   That there is no "Delete" action possible on the flag is reasonable, since the comment is now deleted. However, the flag is/was legitimate, so to decline the flag would be wrong. There's also nothing to leave while editing out the spammy part of the comment, as the comment consists solely of the same spam URL.  There is also no undelete option on the comment, so it's not possible to undelete the comment and then mark the flag as helpful by re-deleting the comment.  Effectively, this means that while the comment itself is effectively gone, the flag on it is in limbo and there is no good choice for what to do with the flag.  When deleting comments normally, my understanding is that any flags associated with those comments are automatically marked as helpful. However, it appears that the deletion of comments when destroying the user who posted them does not do that.  Therefore, I suggest that whenever comments are deleted as part of the owning user being destroyed, any pending flags against those comments should be marked as helpful at the same time.  It would also be a nice touch if those comments showed as deleted by Community♦, instead of Anonymous, but that's less of a concern in my case. Option to hide rep/badges on the avatar block displayed of you Is a general GRE score of X sufficient for entry to postgraduate degree Y? Question: I got general GRE scores of VR (verbal reasoning), QR (quantitative reasoning), and AW (analytical writing).  Is this a good score to apply for entry to postgraduate degree in field X?  How will this score look to the selection committee?    Note: This is an intentionally generic GRE question (with unspecified scores and field of study) which is intended as a template for a large number of questions asked on this forum regarding evaluating the scores on the general GRE test (e.g., , , , , , , , and ).  This question seeks answers that are applicable for an applicant for graduate-level programs in any field, based on any scores in the GRE general test.  The goal of this question is to provide a general reference to avoid the need for specific questions with particular scores/field of study. I have a definition:  \newcommand{\arnold}{Arnold Schwarzenegger}   When I refer to it by:  \arnold is a   it is rendered as:   Arnold Schwarzeneggeris a    In order to have a space in front of "is" I would need to write  \arnold\ is a   Is there some other shorter way? How can a file be copied? How do I copy a file in Python?  I couldn't find anything under . How exactly is Arda supposed to be an ancient Earth? As most Tolkien fans know, Arda (Tolkien's world) is supposed to be Earth in its past, and all its stories are supposed to have happened long ago on this very same planet. Of course, this is fiction, so this is not supposed to be actually true.  But, if one wanted to roll with it... how could it possibly make sense from a geographical standpoint? All maps of Middle Earth (before or after the destruction of Beleriand and Numenor) just don't match anything on Earth, even when considering continental drift.  Of course, one could simply say "the Valar reshaped it (again)", but not only this is a bit unsatisfactory, it also doesn't make much sense in context, as Valar are well known for not wanting to meddle with Arda unless absolutely needed, and even then, as little as possible.    Update:  Hey, please calm down, guys. Mine was not a rant at all. And I know perfectly well this is fiction (and I've stated it plainly). The sense of my question was: "If one wanted to follow this line of thought and assume Arda should be related to Earth as Tolkien said, then how could this relationship be explained?". Difference between == and === in JavaScript What is the difference between == and === in JavaScript?  I have also seen != and !== operators.  Are there more such operators? Terminal icon in window titlebar is stretched out over window control buttons A while ago I upgraded from Ubuntu 17.04 to Ubuntu 17.10 and everything is working fine except for the fact that the icon in the window title bar of my Terminal seems to be stretched out to some ridiculous size.    This only seems to happen in the Terminal window. The Nautilus icon is completely fine.    The problem seems to go away if I login into an Xorg session instead of a Wayland one. I've looked around and didn't find any immediate solution to this. Any help would be greatly appreciated. Where is the dimensions option? Remove Background in Inkscape Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.  For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side of the $x$-axis), whereas a cursory glance would cause one to suspect that adding a positive amount should shift in the positive direction. Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand.  I generally explain this by saying $x$ is getting a "head start". For example, suppose $f(x)$ has a root at $x = 5$. The graph of $f(x+1)$ is getting a unit for free, and so we only need $x = 4$ to get the same output before as before (i.e. a root). Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left.  My explanation seems to help some students and mystify others. I was hoping someone else in the community had an enlightening way to explain these phenomena. Again, I emphasize that the purpose is to strengthen the student's intuition; a rigorous algebraic approach is not what I'm looking for. Derivative of inverse function at a point. Let $f\left(x\right)=x^{13}+3x^{9}+2x^{5}+x^{2}+x+2$ find $\left(f^{-1}\right)'\left(1\right)$. Can somebody show me the method? I am a little confused. no Get More in Plugin Manager in qgis 2 + ubuntu 13.04 After upgrade of qgis 1.8 to qgis 2.0 I couldn't find the choice "Get more" in Plugin Manager and I can't install plugins. What does this notification icon mean? Samsung s7 What is the best practice re: API Version of your Apex Classes? I'm in the process of pushing some updates to some code I originally wrote 12-18 months ago, and the API Version is between 27.0 and 29.0  I know this is to protect the integrity of the test classes you've built (in case they have changed some things) but is there a best practice around either leaving these alone or trying to keep them up-to-date (maybe just once a year instead of at each release)? Is there an algorithm for N body simulations in General Relativity I am new to general relativity but have a background in computer science.  Why is it so hard to do n-body simulations in GR?  For example, there could be a program which takes the properties (mass, position, velocity, etc.) of each particle as input and numerically integrates the evolution of the system using discrete but very small time steps.  If we throw enough memory and computing power at it, we can get arbitrarily close to the 'real' solution.  I assume the entire histories of the worldlines would need to be known, in order to calculate things like gravitational waves.  I have downloaded programs which show the evolution of the Schrodinger equation, so I am surprised they are so hard to find for GR.  Some responses I've heard elsewhere are 'the equations are nonlinear,' but from what I hear numerical algorithms are great for these types of problems.  What types of numerical algorithms are out there for GR? When should I use "a" vs "an"? This question has been answered before in many places but I just need some confirmation with my specific setup. I have Linux Mint installed and I just want a fresh Ubuntu 12.04 install to completely replace Mint. It's dual booted with Windows 7 and I want that to remain completely in tact.  Anyway, I'm currently running the Live CD installer. I've said I want to specify my own partitions and I'm at the following screen:     The list at the bottom expanded:     I confused by all the different partitions. Seems like I should just check the box beside sda5. But what about sda6? Do I just leave that alone?  Also just for confirmation, is the "right" way to replace Mint, or should I be deleting it first? How should I determine how many lands of each different color to put in my deck?