Let $R$ be a commutative ring with $1$. We call $R$ a local ring when $R$ has exactly one maximal ideal $m$. Let $I \subset R$ be a proper ideal of $R$, i.e. $I \neq R$.     Question: Is $I$ contained in $m$?   Approach: I think this is true. If $I$ is a proper ideal of $R$, then $I$ does not contain any units of $R$ (otherwise we would have $I=R$, contradiction!). But this is the only thing I can think of.  Could you please help me solving my question? Thank you!
The statement is:     In a commutative ring with 1, every proper ideal is contained in a maximal ideal.   and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset  A\mid A\text{ is an ideal}\} $, then by set inclusion, every totally ordered subset has a bound, then $P$ has a maximal element $M$.  My question is why $M$ must contain $I$?
Block IPs if they access a resource How does water evaporate if it doesn't boil? When the sun is out after a rain, I can see what appears to be steam rising off a wooden bridge nearby. I'm pretty sure this is water turning into a gas.  However, I thought water had to reach 100 degrees C to be able to turn into a gas.  Is there an edge case, for small amounts of water perhaps, that allows it to evaporate? Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. How to change kernel at boot I installed the latest kernel (4.1) and want to boot into it instead of 3.9. How can i do this on ubuntu 15.04? "In" and "on": How can I decide which one to use for vehicles? Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. How do I write a mathematical equation for ARIMA (2,1,0) x (0,2,2) period 12 Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from. Does anyone know the general strategy for summing a series of the form:  $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$  where $a$ is a positive integer?  Any hints or ideas would be great! Proving an integral equation. When to switch to a CDN? How do I check that multiple keys are in a dict in a single pass? According to some grammars[1][2] and CMS...  "Interjections and vocative should be separated from the rest of the sentence by commas".  The vocative case [1] is used to indicate direct address (i.e., to show when you are talking to somebody or something directly). Nouns in the vocative case are set off using commas. CMS 6.38  As direct address (i.e. vocative case), the parenthetical element should be set off using commas (CMS 6.30). An absolute phrase is always treated as a parenthetical element, as is an interjection.  So,      Hello, world!   That is a very classical way to hello the world.  But, languages change. I am not sure, but today, I feel that it is very uncommon helloing someone with an interjection, providing a small pause and clarifying who is helloed. I have an impression that usually people use hello as a ("kind of") verb, not an interjection.   Question: Is that just a personal (bad) impression or some linguistic has already studied that phenomenon?     [1] - Nelson, Gerald C., Director Survey of English Usage Sidney Greenbaum, and Sidney Greenbaum. An Introduction to English Grammar. Routledge, 2013. [2] - Advanced Grammar in Use with Answers: A Self-Study Reference and Practice Book for Advanced Learners of English. 3 edition. Cambridge University Press, 2013. CMS - Chicago Manual of Style Fixed buffers and interior buffers are not working for a polygon that contains a hole How will the super massive black hole affect our galaxy? How to replace an old version of Ubuntu with a newer one from USB/DVD? I dual booted ubuntu 11.04/11.10 a while ago with a USB stick and i want to uninstall it.   I would like to completely uninstall the old version and get 12.04.  I have a 64 bit laptop. Please help! What is the meaning of speed of light $c$ in $E=mc^2$? is the famous mass-energy equation of Albert Einstein. I know that it tells that mass can be converted to energy and vice versa. I know that $E$ is energy, $m$ is mass of a matter and $c$ is speed of light in vacuum.   What I didn't understood is how we will introduce speed of light?  Atom bomb is made using this principle which converts mass into energy; in that the mass is provided by uranium but where did speed of light comes into play? How can speed of light can be introduced in atom bomb? I want to show that $$f: \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$$ $$x \mapsto x^s$$ is Holder continuous with Holder exponent $s \in \mathbb{R}$, where $0&lt;s \leq 1$. So what I want to show is that $\exists \hspace{2 mm} C \in \mathbb{R}_{\geq0}$ sucht that for all $ x,y \in \mathbb{R}_{\geq0},  $ $$|x^s -y^s| \leq C|x-y|^s$$ and therefore, assuming, wlog $\hspace{1mm} x&gt;y$ $$(x^s -y^s) \leq C(x-y)^s.$$  I thought about Bernoulli's inequality but couldn't make that work.I thought about the binomial theorem, but didn't know how to handle the fact that $s \in \mathbb{R}$. $|f'(x)| \le g(x)$ implies $|f(b) - f(a)| \le \int_a^b g(x) dx$, without assuming $f'$ to be integrable. Why does Beamer not change the itemize symbol when using \setbeamertemplate? This code:  \documentclass[10pt]{beamer} \setbeamertemplate{itemize item}[circle] \setbeamertemplate{itemize subitem}[triangle] \setbeamertemplate{itemize subsubitem}[square]   \begin{document} \begin{frame} \begin{itemize} \item first level     \begin{itemize}     \item second level         \begin{itemize}         \item third level         \end{itemize}     \end{itemize} \end{itemize} \end{frame} \end{document}    produces this output:    I expected the symbol of the first level item to be a circle, not a square. Is there an error in my code? The log file was too long to include in the question, so I . Why didn't Skynet kill Kyle Reese?