So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is?  I believe the definition is:  $I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$  However, after looking about the web, I'm finding that unity has a lot to do with it. I'm struggling to find out how though. I know a unit is an element which can be multiplied by another element in the set (is set the right word?) to get 1.  Essentially I'm asking what it means for a ring to have unity? and what doe an Ideal mean/tell you  I'm just looking for some clarification on this, so any explanation would help, and an example would be great. We were asked to find all Ideals in $Z_6$ today and I didn't understand at the time, so this as an example explained would help me a lot.  Thanks a bunch.
To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these points:   Why is the name "ideal" coined?. In English 'ideal' means "One that is regarded as a standard or model of perfection or excellence." Why did people gave the name of ideal to such group? And why are the ideals not present in the case of groups? And give me a very fantastic intuition and motivation behind the ideals and what are the roles served by them in advanced mathematics.   Thanks a lot, to every one.
Can I wield just a shield and use magic? I play a wizard and while I am rarely hit due to being in the back I'm still paranoid I will get hit. So I was wondering if, after getting a feat to wield a shield, I can use that with one hand and use magic with the other? Composition of two functions - Bijection I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints? How to take the gradient of the quadratic form? Extracting only the BibTex records relevant to the paper My colleague asked me to write a page for a paper he writes. I wrote the page in LyX, compiled it to LaTeX, and sent him the TeX files. Then he reminded me that I forgot to send the .bib file with all the references.  But, my .bib file is huge - it contains over 1000 references (I have a global BiBTeX file for all my papers). The page I wrote uses only about 20 of them.  Is there an automatic way to extract from the bib file, only the records relevant to the current paper? Simplest way to send one-line mail out via command line using gmail? Using Raspbian and Ubunntu 16.04 LTS so need a generic Linux solution.  Requirement is simple:  I need a way to send one-line email messages from the command line.  I have set up a gmail account just for this particular Rpi3, with the address of rpi3abc@gmail.com - with no 2FA  So now I need to be able to send one-line mail messages from anywhere (including cron) without user intervention.    I also would like it to be able to send text files; basically, anything from stdin. Oracle: function based index selective uniqueness No package php-imap available on CentOS 7 I'm running a dedicated server with CentOS 7 64bit, here is the output when trying to install IMAP:    yum install php-imap   Loaded plugins: fastestmirror   Loading mirror speeds from cached hostfile    * base: ftp.ciril.fr    * extras: ftp.ciril.fr    * updates: ftp.ciril.fr   No package php-imap available.   Error: Nothing to do       Also the same when disabling exclusions:    sudo yum install --disableexcludes=all php-imap Loaded plugins: fastestmirror Loading mirror speeds from cached hostfile  * base: ftp.ciril.fr  * extras: ftp.ciril.fr  * updates: ftp.ciril.fr No package php-imap available. Error: Nothing to do What is the difference between "Build and view" and "Compile"? Why does the Hilbert curve fill the whole square? What is the logarithm of a kilometer? Is it a dimensionless number? In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is.  Suppose I have something with length $L = 1 \:\mathrm{km}$.  $\log L = \log \mathrm{km}$  It seems that the unit of $\log L$ is $\log \mathrm{km}$, but I can also say $L = 1000 \mathrm{\:m}$ and now:  $\log L = 3 + \log \mathrm{m}$  This doesn't appear to have any units at all.  This suggests that $\log \mathrm{km}$ and $\log \mathrm{m}$ are actually dimensionless numbers. But wait, I can do this with any unit! Does it actually make sense to talk about the logarithm of a unit, or some other function for that matter? Save all render passes to EXR image? I am having an issue with EXR file export that I did not encounter in previous versions of Blender. I used to be able to render a render layer with all its passes and get all the passes included with the render layer with the EXR export; the problem I am finding now is that I am not able to "pack" the passes with the EXR file (see graphic below). I am working with "standard" EXR, as I understand that the EXR multilayer is for multiple file handling (packing all the image files into a single container), but I may be wrong.    In the screenshot, the graphic is an EXR saved from the rendered layer; you can see in the screenshot how the "extra" passes from the render layer has been ignored by the graphic and only shows the standard image, z and alpha passes. I found this issue for the first time with Blender 2.71; and maybe with Cycles; in the past I have never had an issue with EXR export.  Am I missing something? I hope EXRs still work with Blender, I wouldn't know how to composite without such invaluable aid. Sorry if this question has already been answered somewhere; I have really searched the site for information on EXR but I could not find an answer to my issue. verbatiminput and fontsize I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a single qubit. I'm not sure I completely comprehend this.    Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).   Now I'm not sure if I've understood it right, but since it's still fuzzy in my head it would be great to get some kind of ELI5 explanation or possibly a more detailed picture of how this would, even theoretically, be possible.    Apologies if the question doesn't meet standards. I'm new to the forum and would be open to feedback regarding asking questions or answering them. Capturing speaker output in Java Can I manually limit the %CPU used by a process? How to set shell to wrap words for long commands? Printing pointers in C I was trying to understand something with pointers, so I wrote this code:  #include &lt;stdio.h&gt;  int main(void) {     char s[] = "asd";     char **p = &amp;s;      printf("The value of s is: %p\n", s);     printf("The direction of s is: %p\n", &amp;s);      printf("The value of p is: %p\n", p);     printf("The direction of p is: %p\n", &amp;p);      printf("The direction of s[0] is: %p\n", &amp;s[0]);     printf("The direction of s[1] is: %p\n", &amp;s[1]);     printf("The direction of s[2] is: %p\n", &amp;s[2]);      return 0; }   When compiling it with gcc I get these warnings:  $ gcc main.c -o main-bin -ansi -pedantic -Wall -lm main.c: In function ‘main’: main.c:6: warning: initialization from incompatible pointer type main.c:9: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char (*)[4]’ main.c:11: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char **’ main.c:12: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char ***’   (The flags for gcc are because I must be C89)  Why incompatible types of pointer? Isn't the name of an array a pointer to it's first element? So if s is a pointer to 'a', &amp;s must be a char **, no? And why do I get the other warnings? Do I have to cast the pointers with (void *) in order to print them?   And when running I get something like this:  $ ./main-bin The value of s is: 0xbfb7c860 The direction of s is: 0xbfb7c860 The value of p is: 0xbfb7c860 The direction of p is: 0xbfb7c85c The direction of s[0] is: 0xbfb7c860 The direction of s[1] is: 0xbfb7c861 The direction of s[2] is: 0xbfb7c862   How can the value of s and it's direction (and of course the value of p) be the same? One click shutdown Ubuntu and load into alternative bootup By the "No Hair Theorem", three quantities "define" a black hole; Mass, Angular Momentum, and Charge. The first is easy enough to determine, look at the radius of the event horizon and you can use the Schwarzschild formula to compute the mass. Angular Momentum can be found using the cool little ergosphere Penrose "discovered". However, I don't know how to determine the charge of the black hole.   How can an electromagnetic field escape the event horizon of a Reissner-Nordström black hole? Is there any experiment we could theoretically do to a black hole to determine its charge? prove $\lim_{n\to\infty}{\frac{a^n}{n!}}=0$ I have the proof but i don't understand one part. The proof (for $a&gt;1$) goes as follows:  there exists $k \in N$ such that$a&lt;k, \frac{a}{k}&lt;1$ and since $\frac{a}{k+i}&lt;\frac{a}{k}, i \in N$ $$\frac{a^n}{n!}=\frac{a}{1} \cdot \frac{a}{2} \cdot...\cdot\frac{a}{k}\cdot\frac{a}{k+1}\cdot...\cdot\frac{a}{n}&lt;\frac{a^k}{k!}(\frac{a}{k})^{n-k}=\frac{a^k}{k!}\cdot \frac{a^na^{-k}}{k^nk^{-k}}$$  $$0&lt;\frac{a^n}{n!}&lt;\frac{k^k}{k!}(\frac{a}{k})^n$$  then $$0 \leq \lim_{n\to\infty}{\frac{a^n}{n!}}\le\frac{k^k}{k!}\lim_{n\to\infty}{(\frac{a}{k})^n}$$  Now, the part I don't understand is in the expansion of $\frac{a^n}{n!}$. How does one come with $&lt;\frac{a^k}{k!}(\frac{a}{k})^{n-k}$ part, to be more precise, how does one come up with $(\frac{a}{k})^{n-k}$ part of the inequality?