Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed   subset of $R$ then $S^{-1}R$ is also a principal ideal ring.   Thanks for any insight.
Let $A$ be a commutative ring and $S$ a multiplicative closed subset of $A$. If $A$ is a PID, show that $S^{-1}A$ is a PID.   I've taken an ideal $I$ of $S^{-1}A$ and I've tried to see that is generated by one element; the ideal $I$ has the form $S^{-1}J$ with $J$ an ideal of $A$. $J$ is generated by one element but I can't see why $I$ has to be generated by one element, maybe I'm wrong.
We were taught that an independent clause must have a complete thought/idea.     This is the only mammal that can fly.     How can "This is the only mammal" be treated as an independent clause?            It was a grammar lesson and there was no context there. In this situation how can I decide that it is an independent clause? Though for sure, there should be an independent clause in a correct sentence, and the clause starting with 'that' is not independent. But by this way, I can't fit it with the rule that every independent clause must have a complete thought. A simple sentence must have an independent clause. If the sentence has no independent clause how we can call it a sentence? Can even two dependent clauses form a sentence? Rewrite Sharepoint URL (Sharepoint 2013) I'd like to rewrite all my site urls in the way the the user can't figure out the exact name of the the page his viewing. let's say that my site is "www.justarandomsite.com"  and may page is "page.aspx" I' like to have instead of "www.justarandomsite.com/pages/page.aspx" sothing like this: "www.justarandomsite.com/skdhfjksdgfjksdgfjdsgvdjhsgfjdsdsjfgkdsjgfkdsgfjdsgfjkdsg" When a user on our site loses his password and heads off to the Lost Password page we need to give him a new temporary password. I don't really mind how random this is, or if it matches all the "needed" strong password rules, all I want to do is give them a password that they can change later.  The application is a Web application written in C#. so I was thinking of being mean and going for the easy route of using part of a Guid. i.e.  Guid.NewGuid().ToString("d").Substring(1,8)   Suggesstions? thoughts? What order do triggered abilities happen in if multiple things trigger at the same time? I had several humans on the battlefield.  On turn three, I played . On turn four, I played . On turn five, during my upkeep, this question came up.  At this point, I need to draw/reveal three cards. If I reveal a human, Descendants' Path will allow me to cast it without paying its mana cost.  So, does it matter when I draw the human?  The static abilities state "at the beginning up your upkeep..." which leads me to believe that those two draws/reveals would happen before my normal draw. If that's the case, does the draw order happen in the same order that I played the enchantments?  In the end, that's how we decided to do it. I drew first for Triumph of Ferocity, then I revealed for Descendants' Path, then I drew for my turn. Did we do it right? Editable fields in views "1 out of 100 chickens is" or "1 out of 100 chickens are"? Declaring static data members of normal class and class template Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. Question:      Find the number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. ($f$ is not necessarily unitary, i.e., $f(1)$ need not be $1$.)    Suppose $f$ is a ring homomorphism from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.  Consider $f$ as a additive group homomorphism.  Let $k= |\ker f|$ and  $ t = |\operatorname{im}(f)|$. Then $k\mid 12$ and $t\mid 24$ and $kt=12$, by first isomorphism  theorem of groups.  There are two possibilities $k=3$, $t=4$   and $k=6$, $t=2$.  For the first case $f$ should map $1$ to an element of the subgroup generated by $7$ as there is a unique subgroup of $\mathbb Z_{28}$ of order $4$ generated by $7$. For the second case $1$ has to map to $14$, for the same reasoning.  So there are at most two ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. Question is how to check the possible maps which are ring homomorphisms.  Thanks. Integer solutions for $x^2-y^3 = 23$. As the title stated, I am wondering the integers $x,y$ that satisfy the equation $x^2-y^3 = 23$. “It can be very cold…” vs “It may be very cold…” It can be very cold in winters.   It may be very cold in winters.   What is the difference between these two sentences?  People say when may is used then it means that it may be cold in winters or it may not be cold in winters. In other words, may means 'not sure'.  So does that mean if can is used then it is 100% sure that it will be cold in winters? Is there a way to specify fragile mode as default for all frames? Now I know how to define a fragile frame: \begin{frame}[fragile]. An exercise of Rudin's Real and Complex Analysis says:     Is it true that every compact subset of $\mathbf{R}^1$ is the support of a continuous function? If not, can you describe the class of all compact sets in $\mathbf{R}^1$ which are supports of continuous functions? Is your description valid in other topological spaces?   I proved that, in every metric space, a set (not necessarily compact) $A$ is the support of a continuous function if and only if it is the closure of an open set.  However, I don't think that this characterization holds for every topological space.  So, in order to find a counter-example, I tried to use the discrete and the trivial topologies. However, the discrete topology is metrizable and this characterization holds for topological spaces with the trivial topology.  Is this result true for any topological space? If not, what would be a counter-example? Book: Nanobot apocalypse on Earth, originally used for weather control Can houseplants actually improve air quality? In Python, how do I get a function name as a string, without calling the function?  def my_function():     pass  print get_function_name_as_string(my_function) # my_function is not in quotes   should output "my_function".  Is such function available in Python? If not, any ideas on how to implement get_function_name_as_string, in Python? Automatically run commands over SSH on many servers There is a list of IP addresses in a .txt file, ex.:  1.1.1.1 2.2.2.2 3.3.3.3   Behind every IP address there is a server, and on every server there is an sshd running on port 22. Not every server is in the known_hosts list (on my PC, Ubuntu 10.04 LTS/bash).  How can I run commands on these servers, and collect the output?  Ideally, I'd like to run the commands in parallel on all the servers.  I'll be using public key authentication on all the servers.  Here are some potential pitfalls:   The ssh prompts me to put the given servers ssh key to my known_hosts file. The given commands might return a nonzero exit code, indicating that the output is potentially invalid. I need to recognize that. A connection might fail to be established to a given server, for example because of a network error. There should be a timeout, in case the command runs for longer than expected or the server goes down while running the command.   The servers are AIX/ksh (but I think that doesn't really matter. Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $ I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise... This isn't a particularly major issue, but I've had this thing nagging me for a while. When looking at QGIS 3.4's global variables, user_full_name is assigned my given and surname, which includes a diaeresis (¨) somewhere in there. I can't seem to find any way to get QGIS to switch/adapt encoding and display the name correctly. The accented character gets switched to that ubiquitous black rhombus.  I looked for info on this behavior and found moderately similar issues like  and  but it's not exactly the same problem as it's not a geodata encoding issue but rather how QGIS deals with values pulled from the OS.  I have tried unchecking the Override system locale tickbox in Options|General as I prefer to use American English even though my OS is set to Canadian French. No effect to observe.  There's somewhat of a workaround right now which is to fix the name in the Project variable project_author as it is editable (it's filled with the offending global variable by default).  I'm curious to know, though, if someone has encountered this before and if there's a solution in QGIS. Why were the people of Rhun and Harad allied with Sauron?  Did they join him out of fear, or did Sauron say he would give them something if they joined him?  I know the men of Dunland were angry at Rohan; did Rhun and Harad have something against Gondor? How to install Eclipse?