$\bf{Theorem.}$ Let $R$ be a commutative ring with identity. If $I$ is a proper ideal of $R$, then $I$ is contained in a maximal ideal.   and     $\bf{Zorn's\;Lemma.}$ Suppose a partially ordered set $P$ has the property that every chain in $P$ has an upper bound in $P$. Then the set $P$ contains at least one maximal element.   I have never seen a demonstration of this theorem without the use of the Zorn's Lemma. I know that Zorn's Lemma is a much more general than rings and I don't see a reason for the theorem to imply the Zorn's Lemma, but it makes me wonder if there can be any kind of equivalence between these two things.
Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$.  It's easy to prove that Zorn's lemma implies existence of maximal ideals.  I wonder if the converse proposition is true, i. e., does existence of maximal ideals implies Zorn's lemma? My approach was to construct a ring structure on an ordered set which satisfies Zorn's lemma condition such that existence of maximal ideal implies existence of maximal element in the set, but I didn't succeed.
Do I need to have antivirus software installed? Windows 8 Pro clean install - everything slow; hard drive at constantly at 100% Yesterday, I have downloaded and installed Windows 8 Professional, after downloading it (legally) and burning it to a DVD, on my 3-year-old  (Intel Core Duo 2x2.4&nbsp;GHz, 4&nbsp;GB RAM,  hard disk drive).   The installation failed two or three times, each time differently (error codes, for example 0xc000021a, with an infinite boot loop), but finally I completely formatted the hard drive and installed. Seemingly, everything was OK.  However, after booting successfully, the OS is extremely slow. When I open the task manager, it shows that the HDD is at 100% - however, no application or background process or Windows process can be made responsible. I searched the Internet and found that Windows might try to "index" the hard drive, but I did't find a process that isn't also on my second computer, which still uses the Release Preview (restarting every two hours). Also, the startup is empty, and HD Tune and HDD Health show that the drive health is OK, even though I don't know how reliable these two programs are.  I looked for diagnose software from Toshiba, but they only offer tools for "Fujitsu branded Toshiba devices"; I tried it out, and it did not work on my hard drive.  Does this sound like a hard drive failure? The laptop is 3.5-years-old, has been carried around sometimes roughly and was used (and on) most of the time. However, it would have failed exactly the night before installing Windows&nbsp;8 Profesional, which seems - unlikely.  If this is a software error - then what did I do wrong if this occurs from the moment of installation (even the personalization at the end of installation took half an hour)? How can I fix this? I have two tables in my database:  NEWS table with columns:   id - the news id  user - the user id of the author)   USERS table with columns:   id - the user id   I want to execute this SQL:   SELECT * FROM news JOIN users ON news.user = user.id    When I get the results in PHP I would like to get associative array and get column names by $row['column-name']. How do I get the news ID and the user ID, having the same column name? showing that a group of order 45 is abelian What does "Package  has no installation candidate" mean? What is the MIME equivalent of uuencode for sendmail? e.g., I've got files I want to send by sendmail but the receiving mailbox (for posting pictures to my blog) will not accept uuencode files a la   $uuencode file | sendmail user@domain  I'm guessing it'll accept MIME though. So what command can I use instead? Floats appearing in wrong places Is there a way to install a newer version than in the repository?  Ubuntu installs vagrant 1.4.3 at the moment, but I need at least 1.5 How to adapt classicthesis layout to B5paper and Linux Libertine Question says it all: I am using classicthesis with XeTeX to typeset my PhD thesis. I want to use B5 paper size (176 × 250 mm) and the beautiful Linux Libertine font. I'm wondering how to    calculate the optimal page margins actually apply these settings to the document   Any help is greatly appreciated =)  EDIT: I should add that I need to generate two different PDF files from this:   a version with papersize A4 (210 × 297 mm), where the B5 layout is (approximately) centered, and crop marks are shown, to pass on to the printing company a version with papersize B5 for online distribution, having the same layout as the printed one.   I tried fiddling with using  and \areaset command, but to no avail. This question led to a new package:      I'm trying to make some good-looking numbers in a circle, using the simple command:  \textcircled{1}   However, the circle is misaligned vertically with the number, and look a bit sloppy:    Any suggestions how can I adjust the vertical alignment so that the numbers look more centered w.r.t to the circles? Or, if that's not an option, what other ways to achieve the same results are possible?  I will use those only for numbers, and in very few places, so manual adjustment per glyph is an option.  I'm using pdfLaTeX with Computer Modern, if that matters. Mid-2017  started publishing monthly updates. This post will continue that effort into 2018. (.)  Updates are posted at the beginning of the month. They cover what was done last month, what is still in progress and new work that will be started in the current month.  Updates:      (CoC, nav, OpenID, responsive design, bugs, campaigns)  (CoC, comment flags, nav, OpenID)  (question wizard, nav, user survey, OpenID)  (tag watching, GDPR)  (responsive design, tags, teams)  (responsive design, OpenID, teams, sponsorship)  (responsive design, facebook auth, question lists, teams)  (newsletter, sign up, Documentation reputation, channels)  (top bar issues, hats, question template) For example:  me$ FOO="BAR * BAR" me$ echo $FOO BAR file1 file2 file3 file4 BAR   and using the \ escape character:  me$ FOO="BAR \* BAR" me$ echo $FOO BAR \* BAR   I'm obviously doing something stupid.  How do I get the output BAR * BAR? Convergence of integrals in $L^p$ Stuck with this problem from Zgymund's book.   Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1&lt;p&lt;\infty$. Assume that $\|f_{n}\|_{p} \leq M &lt; \infty$. Prove that:  $\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac{1}{q} = 1$.  Right, so I estimate the difference of the integrals and using Hölder end up with:  $$\left|\int f_{n} g - \int fg\right| \leq \|g\|_{q} \|f_{n} - f\|_{p}$$  From here I'm stuck because we are not assuming convergence in the seminorm but just pointwise convergence almost everywhere. How to proceed? Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) Itemize: Identation is different between first and second line What is potential energy truly? Agreement With Compound Subjects Joined by And I was surfing the internet the other day when I found this phrase: Instead your precious time and attention is wasted. To my ears, it sounds wrong. But I'm not a native English speaker, so I consulted with a number of them, and some of them said that it was "is" because the adjective modifies only the noun "time" and not "attention". Others said it "is" because that's just how it is. But I still believe it should be "are". Any help is appreciated. Crash course on algebraic geometry with view to applications in physics Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?   What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references. Convert Month Number to Month Name Function in SQL I have months stored in SQL Server as 1,2,3,4,...12. I would like to display them as January,February etc. Is there a function in SQL Server like MonthName(1) = January? I am trying to avoid a CASE statement, if possible. How do we know the brain flips images projected on the retina back around? Why do we turn images upside down again rather than dealing with them directly, still vertically rotated after passing through our lens?   I don't see how that would cause any problems, and how we'd ever be able to figure out if we are presented with flipped images after getting used to interacting based on visual input, whether flipped or not.   What am I missing?