Suppose that $S\subset R$ is a multiplicative set in $R$, where $R$ is a commutative ring with identity $1\neq 0$. Prove that the homomorphism $\phi:R\to S^{-1}R$ is injective if and only if $S$ contains no zero-divisors.  I have an idea about how to approach this question, but I am not sure exactly how to structure it. For one direction, where we assume $\phi$ is injective, this means that $\ker(\phi)=\{0\}$, which I think means that there are no zero divisors in $R$, so there are no zero divisors in $S$ because it is a subset of $R$? Am I headed in the right direction?  Edit: I believe this question is different than the one it is marked as duplicate to, because that question uses the concept of monomorphisms which are not mentioned here.
Let the ring of fractions be denoted as $S^{-1}R$.       Proposition (from Robert Ash's textbook "Basic Abstract Algebra")      Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring homomorphism. If S has no zero divisors then f is a monomorphism, and we say that R can be embedded in $S^{-1}R$. In particular:      (i) A commutative ring R cna be embedded in its complete (or *full) ring of fractions ($S^{-1}R$, where S consists of all non-divisors of zero in R).       (ii) An integral domain can be embedded in its quotient field.    Why does the proposition say "If S has no zero divisors..."? If S really had zero divisors, then this wouldn't even be properly defined, right? Because we would have a/0 for some a in R.
No, this is not a , nor is it a solicitation on  to backup or  and only a little bit about Where to backup and When to do it.  This is basically a strategy question: what, where and when all together. "How" is not really relevant: there are tons of back-up programs out there and they all do the same basic thing: make copies of data.  But the real question is: are these back-up programs making copies of the data important to you?  How should you install&amp;run Ubuntu while safeguarding your data?  There is no "one size fits all" when it comes to computers, Ubuntu versions or a backup strategy, therefore I've split this up into 5 basic user types with each their own answer:       (or you're running in UEFI mode!)   And as the above does not fit everyone, one for the impatient:   !    So click one of the above links and go to your user type!  P.S. If you're running a server farm and you've read this question so far, I'm sorry: this is about a single PC, not a network, nor a server farm. Linear regression prediction interval error in running bash script as root I have a simple shutdown script which i want to run as root  shutdown -h +30; echo "succesfull";   I have followed these answer, to make my script root.  first I run this command  sudo chown root.root $HOME/test/test.sh sudo chmod 4755 $HOME/test/test.sh   and then made changes in sudoers  sudo visudo   added this command after this line %sudo   ALL=(ALL:ALL) ALL  eka ALL=(ALL) NOPASSWD: $HOME/test/test.sh   But when i executed my script its showing this error  shutdown: Need to be root succesfull How to reshape data from long to wide format The pdf of $\frac{X_1-\bar{X}}{S}$ Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2&gt;0$  Let $Z=\frac{X_1-\bar{X}}{S},$ S is the standard deviation here.  It is can be shown that  $Z$ has the Lebesgue p.d.f.  $$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$$  My question is then how to get this pdf?  The question is from  in example 3.3.4  to find the UMVUE of $P(X_1 \le c)$. I can understand the logic and procedures to find the UMVUE but don't know how to get the pdf.  I think this question also relate to this    Thank you very much for help or point to any related references will be also appropriated. Is there anyway I can connect resistors to allow them to take more power such as using 4 1/4 watt resistors to get a 1 watt resistors. Or do i just have to use 1 1 Watt resistor. Does the following limit exist? $$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$  If yes, then I have to find it; if no, then I need to give reason why. I couldn’t figure out how to prove this formally, can someone please help me out. I can understand this requirement for the old PPC RISC systems and even for x86-64, but for the old tried-and-true x86? In this case, the stack needs to be aligned on 4 byte boundaries only. Yes, some of the MMX/SSE instructions require 16byte alignments, but if that is a requirement of the callee, then it should ensure the alignments are correct. Why burden every caller with this extra requirement? This can actually cause some drops in performance because every call-site must manage this requirement. Am I missing something? Update: After some more investigation into this and some consultation with some internal colleagues, I have some theories about this:  Consistency between the PPC, x86, and x64 version of the OS It seems that the GCC codegen now consistently does a sub esp,xxx and then &quot;mov&quot;s the data onto the stack rather than simply doing a &quot;push&quot; instruction. This could actually be faster on some hardware. While this does complicate the call sites a little, there is very little extra overhead when using the default &quot;cdecl&quot; convention where the caller cleans up the stack.  The issue I have with the last item, is that for calling conventions that rely on the callee cleaning the stack, the above requirements really &quot;uglifies&quot; the codegen. For instance, what some compiler decided to implement a faster register-based calling style for its own internal use (ie any code that isn't intended to be called from other languages or sources)? This stack-alignment thing could negate some of the performance gains achieved by passing some parameters in registers. Update: So far the only real answers have been consistency, but to me that's a bit too easy of an answer. I have well over 20 years experience with the x86 architecture and if consistency, not performance, or something else concrete, is really the reason then I respectfully suggest that is a bit naive for the developers to require it. They're ignoring nearly three decades of tools and support. Especially if they're expecting tools vendors to quickly and easily adapt their tools for their platform (maybe not... it is Apple...) without having to jump through several seemingly unnecessary hoops. I'll give this topic another day or so then close it... Related How to download/read only the first 80KB of a file? $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ?? I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:  $V(1): 1≤1 \text{  true}$  $V(n): n!≤((n+1)/2)^n$  $V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$  and I've got : $(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ $((n+1)^n)n(n+1)≤((n+2)^n)((n/2)+1)$ Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence Would someone please advance/discuss some real-life situations falsities $1, 2$? I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I haven't naturalised these concepts yet. Thus, I beg leave to ask about other real-life examples which require less or no computations.  I tried and would appreciate less numerical examples than  and , and  Examples 1.22 and 1.23 on P39 of Introduction to Pr by Bertsekas.  $1.$ Pairwise Independence $\require{cancel} \cancel{\implies}$ (Mutual) Independence.      $2.$ Pairwise Independence $\require{cancel} \cancel{\Longleftarrow}$ (Mutual) Independence.         P38 defines (Mutual) Independence :    For all $S \subseteq \{1, ..., n - 1, n\} $ and events $A_i$, $Pr(\cap_{i \in S} A_i) = \Pi_{i \in S} A_i.$ I would like to replace the usual \hat in maths with a larger one, but I do not want it to stretch. The size I want is "the smallest \widehat" (as in $\widehat{.}$). Is there a way to put that one above a wide character like M ($\widehat{M}$ would stretch and give be an even wider hat).  I guess I have a lack of understanding for the mechanism behind accents that extend, so feel free to explain. When to use a preceding slash in path names? (e.g. for the 'cd' command) I am a novice user and I am trying to install Oracle Java. The process calls for me to access /usr/local/java, but here is the problem. I get as far as /usr/local, then this is the message I get:  leonard@leonard-MT6452:/usr/local$ dir bin  etc  games  include  java  lib  man  sbin  share  src  leonard@leonard-MT6452:/usr/local$ cd /java/ bash: cd: /java/: No such file or directory  leonard@leonard-MT6452:/usr/local$ rm -r ~/java rm: cannot remove ‘/home/leonard/java’: No such file or directory   Help? Not able to add 3 JPanels to a main panel How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$ Test if there are files matching a pattern in order to execute a script Distributing distinct apples and pears to three people so that each person receives at least one pear How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear.   Note: The first part of the problem was the same but the word distinct was replaced with identical apples and pears. That answer is $360$ ways.   What has to be done differently when the word identical is replaced with distinct? Will I miss anything by watching Dragon Ball Z Kai instead of the other series? I've read that Dragon Ball Z Kai is an HD remaster and recut of Dragon Ball and Dragon Ball Z. However, Kai has  while Dragon Ball has  and Z has .  If I watch only Dragon Ball Z Kai, do I miss anything I would get by watching the other two series? And is anything censored in Kai that is not censored in the others? Spoilers for A Storm of Swords / Season 4 episode 2 "The Lion and the Rose".  At the Purple Wedding,      Littlefinger and Lady Olenna Tyrell   perform their plan of      poisoning King Joffrey.   A short summary of their plan:  Littlefinger orders Ser Dontos Hollard to give Sansa Stark a beautiful silver hairnet adorned with amethysts. At least some of the amethysts are actually strangler crystals in disguise. Ser Dontos also, still under orders of Littlefinger, urges Sansa to wear the hairnet at the wedding, which she then does. At the wedding, when Lady Olenna greets Sansa, she fusses her hairnet a bit, while secretly taking one of the strangler crystals. Later, during the wedding, Lady Olenna puts the crystal in Joffrey's royal cup, poisoning him.  But why was their plan so complicated? Why couldn't Lady Olenna just bring a strangler crystal with her to the wedding (as suggested in )?  Then they wouldn't depend on Ser Dontos and Sansa who could unknowingly have done something to prevent the success of the plan.  The only reason that I could think of, is Littlefinger and Lady Olenna plan for Sansa (and Tyrion as her husband with her) to take the blame for Joffrey's death.  They put the crystals on her person, so that upon examination it would clearly show that Sansa had the murder weapon on her. But Littlefinger had a plan he did not tell Lady Olenna, namely rescuing Sansa during the chaos ensuing the murder (and thereby leaving Tyrion behind as the only accused).  Is there anything confirming or debunking this theory? If my theory doesn't hold up, is there another explanation for why the plan was so needlessly complicated?