In ZF without choice, can a commutative ring $R$ have a prime ideal $P$ that is maximal among prime ideals but not among proper ideals?   By replacing $R$ with the quotient ring $R/P$, we may assume WLOG that $P$ is the zero ideal, so that $R$ would then be an integral domain that is not a field but has the zero ideal as its only prime ideal.  Note that it is possible for an integral domain (in ZFC or otherwise) to have exactly one nonzero prime ideal (e.g. any discrete valuation ring).
I would like to solve the beginner's standard exercise which claims that a point of $\mathrm{Spec} \ R$ is closed iff it is a maximal ideal.  The reverse implication is easy. The direct one seems much subtler. If $P = V(I)$ then, since there exist a maximal ideal $M$ containing $P$, it would follow that $\{P, M\} \subseteq V(I) = P$, whence $P=M$. In order to show the existence of $M$ I applied Zorn's lemma to the set of ideals containing $P$.  The question is the following: is there any other proof of the above implication that does not (indirectly) use the axiom of choice? My concern is that I may be using a cannon to shoot a fly. Also, it would be the first time that I see closedness of points to require the axiom of choice. If the axiom is indeed needed in general, are there "nice" classes of rings for which we could get away without it?
Before I met her I had a poor opinion of her.   As person had a poor opinion before a particular time in past (before He met her) so I think there should be had had.   There are some more example like this:   Before I went to the university, I worked/had worked as a carpenter for 5 years. Before I opened the door, I put/had put the key in the lock. Before I opened the door, I looked/had looked carefully into the barn for snipers.   In all these sentence, for me, it should be perfect tense.  I asked that question on another platform where I was told that if sequence of past event is obvious and sentences uses "before"and "after", we often use past tense. Which means we should use past tense in all these sentences. But I have another sentence "The train had left before I reached the station." If I follow above mentioned rule there should be "left" instead of "had left". But I am damn sure this sentence is correct with "had left".   Am I correct?  Thank you Find the limit $$\lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$$  My attempt: Since $$\sin{x}=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\dfrac{1}{7!}x^7+o(x^7)$$ $$\tan{x}=x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+\dfrac{1}{63}x^7+o(x^3)$$ So $$\sin{(\tan{x})}=\tan{x}-\dfrac{1}{3!}(\tan{x})^3+\dfrac{1}{5!}(\tan{x})^5-\dfrac{1}{7!}(\tan{x})^7+o(x^7)$$ Though this method might solve, I think this problem has nicer methods. Thanks. Most textbooks introduce differentiable manifolds via . This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.  There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via . This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:   A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.   This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:   The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read). Somewhere in that definition, the condition that $M$ be paracompact is implicit.   Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?    Also, is this definition given carefully in any textbook?  Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative. How can I escape special HTML characters in JSP? How to repair a broken .EXE file association $x^2+y^2=z^n$: Find solutions without Pythagoras! I was presented with the following problem:      Prove that there exist solutions to $x^2+y^2=z^n$ for all $n$, with $x,y,z, n \in \mathbb{N}$   I showed that by taking any Pythagorean triple $x^2+y^2=z^2$ and multiplying by $z^{2(n-1)}$ we get $(z^{n-1}x)^2+(z^{n-1}y)^2=(z^2)^n$, which allows me to generate solutions easily for any value of $n$. I managed to find several similar questions on this site, such as  concerning the specific case $n=3$. I notice that all of these questions take a similar approach and start with a Pythagorean triple and use it to generate general solutions.  Is there a way to prove the statement (or better yet provide solutions to the equation) without first relying on Pythagoras? I have always wondered why diamonds shine. Can anyone tell me why? How to open and convert CHM documents? I have some documents that are in a .chm format. I wondered if there is a file format that can be easier to navigate, supported and of equal file size in Ubuntu?  If there is, I would like to start converting all those books and probably using them with less hassle on all my Ubuntu PCs and my Android phone. Why is integration so much harder than differentiation? How do I programmatically obtain the version in JBoss AS 5.1? What is the difference between Newtonian and Lagrangian mechanics in a nutshell? Can a multi-core processor run multiple processes at the same time? I understand that a multi-core processor can run multiple threads of the same process at the same time using its different cores. Can it also run multiple processes at the same time(each core running a different process)? How to automatically generate a stacktrace when my program crashes I am working on Linux with the GCC compiler. When my C++ program crashes I would like it to automatically generate a stacktrace.  My program is being run by many different users and it also runs on Linux, Windows and Macintosh (all versions are compiled using gcc).  I would like my program to be able to generate a stack trace when it crashes and the next time the user runs it, it will ask them if it is ok to send the stack trace to me so I can track down the problem. I can handle the sending the info to me but I don't know how to generate the trace string. Any ideas? How can one format a cell to have word wrap in Google Sheets?  I've tried Shift+Enter, but that doesn't seem to work.  One thing I know works is to place  and then if you edit your cell you can have word wrap in it.  But is there a way to do this without switching modes? Prove that $(a+1)(a+2)...(a+b)$ is divisible by $b!$ The problem is following, prove that:  $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$    I've tried solving this problem using mathematical induction, but I don't think that i did it correctly. Here's what i've done  $1.\ b=1\ (Basis)$  $$b!| (a+1)(a+2)...(a+b)$$ $$1 | a+1 \text{, which is true}$$  $2.\ b=k\ (Induction\ Hypothesis)$  $$k! | (a+1)(a+2)...(a+k)\text{, we assume it's true}$$ $$k!*n = (a+1)(a+2)...(a+k)\text{, n is some positive integer}$$  $3.\ b=k+1\ (Inductive\ Step)$  In order to prove I should get:  $$(k+1)!*m = (a+1)(a+2)...(a+k)(a+k+1)\text{, where m is some positive integer}$$ $$(a+1)(a+2)...(a+k)(a+k+1) = k!*n*(a+k+1)$$ $$(a+1)(a+2)...(a+k)(a+k+1) = (k+1)!*n + k!*n*a$$ $$(a+1)(a+2)...(a+k)(a+k+1) - a(a+1)(a+2)...(a+k) = (k+1)!*n$$ $$(a+1)(a+2)...(a+k)(k+1) = (k+1)!*n$$  And as you can see I'm returning at the beginning.  Also I've tried to use the fact that in b consecutive numbers, there must be at least one that divides b, but since i eliminate that number (because I couldn't be sure that the quotient is divisible with (b-1) or (b-2) or ... or 2. And after this i can't continue with (b-1) using this method, because it's not necessary for the other (b-1) to be consecutive. In how many ways can we pick three different numbers out of the group $1,2,3,\dots,100$ such that the largest number is larger than the product of the two smaller ones? (The order in which we pick the numbers does not matter.) Difference beween wiki page, application page, web part page and publishing page There are lots places in msdn sharepoint forums and blogs where these terms are used and I am confused between them. Can anybody clarify? difference fn(String... args) vs fn(String[] args) Whats this syntax useful for :       function(String... args)   Is this same as writing       function(String[] args)    with difference only while invoking this method or is there any other feature involved with it ? Is there any IC that interfaces any 3.3V Input/Output to 5V Output/Input? I need it primarily for an Arduino Due but if there any bidirectional IC that works like that it would be great.  Some people advised me to use the SN74AHC125 and CD4050 ICs, but I don't understand how they work or how to interface with them. Plotting Population Regression Function with pgfplot I want the make the following population regression function with pgfplot.     My attempted code is  \documentclass[border=5mm]{standalone} \usepackage{pgfplots}   \begin{document}  \begin{tikzpicture}[ % Define Normal Probability Function declare function={             normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));         }        ] \begin{axis}[     no markers   , domain=-3.2:3.2   , samples=100   , ymin=0   , axis lines*=left   , xlabel=     , every axis y label/.style={at=(current axis.above origin),anchor=south}   , every axis x label/.style={at=(current axis.right of origin),anchor=west}   , height=5cm   , width=12cm   , xtick=\empty   , ytick=\empty   , enlargelimits=false   , clip=false   , axis on top   , grid = major   , hide x axis   , hide y axis   ]    \addplot[cyan!50!black, rotate=-45] (x, {normal(0, 1)});  \end{axis}   \end{tikzpicture}  \end{document}     Any help to get this graph will be highly appreciated. Thanks