Let $R$ a commutative ring (with $1$) such that $x^2=x$ for all $x\in R$.  Prove that for every $x,y\in R$ exist $z\in R$ such that $\langle x,y\rangle=\langle z\rangle$  Any idea with this? the exercise had an item a) that said to prove that $2x=0$ for all $x\in R$, but I can't see how to use it.
The classical book on commutative algebra Introduction to Commutative Algebra, by Atiyah and Macdonald, has the following as exercise I.11.       A ring is Boolean if $x^2=x$ for any $x$ of $A$. In a Boolean ring $A$, show that     i) $2x=0$ for all $x\in A$;    ii) Every prime ideal of $A$ is maximal, and its residue field consists of two elements;   iii) Every finitely generated ideal of $A$ is principal.     I can but solve the first two of them; for the last one, I cannot even perceive the situation here, for it will not, generally speaking, be a Dedekind domain, whereof my knowledge is somewhat more. I have hitherto tried to find one element for each ideal of two generators which can stand the stead of them, and then proceed by induction. But even the first step fails to produce any result. Thanks for any hints and inspirations in advance.
Group presentation for discrete Heisenberg group Which package can be used to write BNF grammars? Enable HTML5 tags on Eclipse I really love Eclipse (PDT)... But i'm developing sites using HTML5 and CSS3 and it mark the HTML5 tags like &lt;nav&gt;, &lt;section&gt; and &lt;article&gt;  as invalid tags (inexistent).  How can I add those new tags to it?  And what about CSS3 rules?  I read something about remove the "invalid tag" check... and that's not what I want. :( How to get the current directory in a C program? I'm making a C program where I need to get the directory that the program is started from.  This program is written for UNIX computers. I've been looking at opendir() and telldir(), but telldir() returns a off_t (long int), so it really doesn't help me.   How can I get the current path in a string (char array)? GridView sorting: SortDirection always Ascending How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$? Possible Duplicate:          How can we integrate $$\int_0^\frac{\pi}2\frac{\sin^nx}{\sin^nx+\cos^nx}\,\mathrm dx , \,\,\,\,\,\,\,\,\, n\in N \quad?$$ Thanks for any hint. Cropping the exact same part of several pictures I have several pictures that are exactly the same except for some small variations. It was one picture that was photoshopped in several different ways. Now, I need to crop them all (13 in total) but I need to crop exactly the same part of it. It needs to be pixel perfect.  How do I do that with Pixelmator? Per-chapter bibliographies in biblatex I'm really new in using biblatex, compile the sample placed in , but what I get is    1.1 Foo    Some text [B02x ].    1.2 Bar    Some text [B02y ].   Do not get the bibliography by chapters, or the global references.  Really not remains to be done   \documentclass{report}  \usepackage[style=alphabetic]{biblatex}  \addbibresource{references.bib}  \begin{document}  \chapter{First chapter}  \section{Foo}  Some text \cite{childs_temperature}.  \printbibliography  \chapter{Second chapter}  \section{Bar}  Some text \cite{hashemian}.  \printbibliography  \newpage  \printbibliography  \end{document}   And the references.bib    @BOOK{childs_temperature,   title = {Practical Temperature Measurement},   publisher = {Butterworth - Heinemann},   year = {2001},   author = {Childs, Peter R N},   address = {Great Britain},   edition = {1},   isbn = {0 7506 5080 X}   }    @PHDTHESIS{hashemian,   author = {Hashemian, Hashem Mehrdad},   title = {Measurements of dynamic temperatures and pressures in nuclear power plants},   school = {{The University of Western Ontario}},   year = {2011},   type = {PhD {T}hesis}   }   I need to obtain references by chapter and global cited references, but i don't know how compile. How to automount a partition on login? I have my musics in another partition of the HD, and I would like that, on login, the partition to be auto-mounted, so I can just open the media player and listen to my music. Is there a way? After S.H.I.E.L.D confiscates Jane's work in Thor, the researchers discuss what they'll do next.  Erik mentions two colleagues. The first is reasonably implied to be Bruce Banner, but who is the second colleague he mentions? Transcribed from the film:  Jane Who are these people? Erik I knew this scientist... a pioneer in gamma radiation. S.H.I.E.L.D showed up and... umm. He wasn't heard from again. Jane They're not going to do that to us. I'm going to get everything back. Erik mumbles Please. Let me contact one of my colleagues. He's had some dealings with these people before. I'll email him and maybe he can help. Darcy They took your laptop, too. Does a car use friction to move? When a car's engine injects fuel into the cylinder chambers, the reaction creates a force that generates rotational momentum to the shaft and over the transmission, it translates that power to the wheels, right?  But, something's bothering me, how does the car actually move, is it dependent on the force of friction? Wheels have a contact with the road in one exact point and when the drive gets to rotate them, a force is exerted on the road in the direction of movement and there is friction that is counteracting it.  Is the key to get a car properly moving to have the drive force equal or less than the force of friction to keep that point locked to the ground and use the rolling of the wheel for translational displacement ie. moving forward? If it's bigger than the force of friction, that would cause the wheels to spin in place?  Am I completely off or am I getting something right? If there's a knowledgeable person on this topic, I'd greatly appreciate some insight, perhaps even a bigger response with some basic vehicle physics. Laptop w/o battery US bound flight? During kill Bill, we learn that all the black mamba had to do to get Hanzo to build her a sword, was to mention Bill's name.  When black mamba meets Bill at the end, she tells him that all she did was mention his name. Bill responds with, "that'll do it!"  So what was it that Bill did to Hanzo to make him come out of retirement and make another weapon of death? How do I join my partitions? a better way to express "an idea/thought suddenly came to me" What are some grandiloquent, or simply better, ways of expressing "an idea/thought suddenly came to me", or "an idea/thought struck me", or "I was struck by an idea/thought"? Autokey (autokey-qt) freezes and won't insert text I'm having some trouble understanding a proof on  on Number Theory and I was wondering if you guys could give me some help. The problem is that I don't understand the last implication on the proof for example 1.1     Example 1.1. Let x and y be integers. Prove that 2x + 3y is divisible by 17 iff 9x + 5y is divisible by 17.      Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒ 17 | (9x + 5y)       conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or 17 | (36x + 20y) ⇒ 17 | (2x + 3y).   My problem is that I don't understand how does 17 | (26x + 39y) imply 17 | (9x + 5y). If you could elaborate on this step I would be most grateful.  I'm sorry if this is an obvious question but I am a beginner and I just can't get it.  Thanks for your help in advance. I have a pretty simple table that needs to span multiple pages. This seems like a job for either the / package or the  package.   The voting on the answers to this  suggest that longtable is the preferred package, yet xtab is much newer and has been revised since that question was asked.  What are the differences between xtab and longtable and when is one preferable to the other? Cannot edit post meta fields with rest API I am using python to send POST requests to /wp-json/wp/v2/&lt; post type &gt;/&lt; post id &gt;. This successfully updates standard fields such as title, content etc. However I am unable to make any changes to the custom theme meta fields structured like &quot;post_meta&quot;:{     &quot;meta_1&quot;:[],      .      .      &quot;meta_n&quot;:[] }.  When I make a GET request to the same endpoint, the post meta fields are returned so I don't think this is an issue with the fields not being registered? Thanks Number of points on a line in a finite projective plane I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for finite projective planes? Or does this relate to the order of the finite field being projected? (for example in the Fano plane, does the fact that each line contains 3 points follow from the fact that there are 3 elements {0,1,2} in Z_2?) I haven't been able to find a clear explanation of this (clearly I do not yet have a solid understanding of how such planes are constructed). If someone could enlighten me that would be excellent.