Atyiah-Macdonald Ex 3.7 A multiplicatively closed subset $S$ of a ring $ A$ is said to be saturated if $xy\in S$ $\iff$ $x\in S$ and $y\in S$. Show that if $S$ is saturated if and only if  $A-S$ is a union of a primes.  First of all I assume they mean the union where there are no redundant ideals in the union. For example in $\mathbb Z$, $(2)\cup (3)\cup (6)=(2)\cup (3)$ and this statement cannot be both true and false. Lets work assuming nonredundant union is used. If $A-S$ is a union of primes it follows easily that $S$ is saturated. Let $S$ be saturated. Then certainly $A-S$ is a union of primes as it it is the union of each $(x)$ with $x\notin S$. We would like to know that we can write this as a union of prime ideals. Since we know we can write $A-S$ as a union of ideals, I think we can assume that we have written as a union of ideals in a non-redundant way. We do not know that these are prime. Suppose that one of the ideals $I$ in the union is not prime. Then there is $xy\in I$ such that neither $x$ or $y$ are in $I$. We cannot have $x$ and $y$ in $S$ since $xy\notin S$. Therefore $x$ must be in some other ideal $J$ in the union. This does not seem to give us enough to contradict the union being non-redundant. Is it possible to continue with this approach or how should I tackle this?
Complement of maximal multiplicative set is a prime ideal Let $R$ be a commutative ring with identity. I've been trying to prove the following:     If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$.   I have spent some time in it, but nothing is coming to my mind. Any idea/solution will be appreciated.
"Eat" is to "feed" as "drink" is to what? Does the Rule of Two Serve any Actual Purpose? I know Darth Bane came up with the Rule of Two to ensure the survival of the Sith, but, in reality (or in the Star Wars reality), does it (or any Sith rule) serve a true purpose?  And why does each successive Sith bother with it?  Due to the nature of the Sith, Sith Lords tend toward arrogance and self-importance.  Both the Sith master and Sith apprentice seem to regularly break the rule by training others.  Deceit and personal interest over loyalty is a way of life for the Sith.  What is to keep any Sith from following the Rule of Two (or any rule, for that matter) if it is against their own self interest?  If a Sith Master can stay in power longer by training a 2nd apprentice and pitting the two against each other, is he really going to worry about the Rule of Two more than his own plans and survival?  The Rule of Two also assumes that every Sith Master would place the survival of the Sith over his own survival, as opposed to attempting to ensure his own immortality in whatever way he could (like Plagueis).  When you have a group that, by default, is arrogant and self-serving, as well as deceptive, why would they want the Sith to survive themselves?  I can see how the Rule of Two, and any other Sith beliefs would be possible guidelines, but it terms of the nature of ambitious people (and the Sith are ambitious), it's hard to believe that each new Sith Master would follow this rule faithfully.  So does this rule, or any Sith rule, actually work?  Or is it more of a suggestion that is broken when convenient? "PuTTY key format too new" when using ppk file for PuTTY SSH key authentication I wanted to connect to my Unix server using SSH keys. For that I had converted my id_rsa file from ./ssh directory to a ppk file using PuTTYgen, but after loading my .ppk key I get this warning:  PuTTY key format too new   If I use my default id_rsa file, then I get error:  OpenSSH SSH-2 private key (old PEM format)  Either way I am getting stuck. Can anyone please help me? Can't install Skype 4.3 on Ubuntu 14.04 64 bit First of all, I think this question is not duplicated, because I have tried all other solutions I found here, but nothing solved my problem.  I use Ubuntu 14.04 64 bit.  I have tried until now: sudo apt-get install skype -> It didn't work. The error was:   The following packages have unmet dependencies: skype : Depends: skype-bin E: Unable to correct problems, you have held broken packages.   I have tried sudo dpkg add-architecture &amp;&amp; sudo dpkg -i skype-ubuntu-precise_4.3.0.37-1_i386.deb but it didn't work either. The error was:  dpkg: dependency problems prevent configuration of skype: skype depends on libqt4-dbus (&gt;= 4:4.5.3); however: Package libqt4-dbus is not installed. skype depends on libqt4-network (&gt;= 4:4.8.0); however: Package libqt4-network:i386 is not installed. skype depends on libqt4-xml (&gt;= 4:4.5.3); however: Package libqt4-xml:i386 is not installed. skype depends on libqtcore4 (&gt;= 4:4.7.0~beta1); however: Package libqtcore4:i386 is not installed. skype depends on libqtgui4 (&gt;= 4:4.8.0); however: Package libqtgui4:i386 is not installed. skype depends on libqtwebkit4 (&gt;= 2.2~2011week36); however: skype depends on libstdc++6 (&gt;= 4.2.1); however: Package libstdc++6:i386 is not installed. skype depends on libx11-6; however: Package libx11-6:i386 is not installed. skype depends on libxext6; however: Package libxext6:i386 is not installed. skype depends on libxss1; however: Package libxss1:i386 is not installed. skype depends on libxv1; however: Package libxv1:i386 is not installed. skype depends on dpkg: error processing package skype (--install): dependency problems - leaving unconfigured Processing triggers for hicolor-icon-theme (0.13-1) ... Processing triggers for gnome-menus (3.10.1-0ubuntu2) ... Processing triggers for desktop-file-utils (0.22-1ubuntu1) ... Processing triggers for bamfdaemon (0.5.1+14.04.20140409-0ubuntu1) ... Rebuilding /usr/share/applications/bamf-2.index... Processing triggers for mime-support (3.54ubuntu1) ... Errors were encountered while processing: skype   I have tried to use after sudo apt-get -f install but it didn't solve the issue.  I have tried to install by using Gdebi. The error was:  Cannot install 'libqtwebkit4:i386'   I have tried to install using Ubuntu Software Center. The error was:  Cannot install 'libqtwebkit4:i386'   I have tried changing the server in "Software and Updates". I have tried to use "Main Server" and "Servers for United States". But it couldn't solve my problem.  Anyone has any idea how to solve that?  SOLVED  Well, I formatted my pc and I installed Skype first of all. Probably another installation that I had done before the Skype installation was the problem, i guess.  Thanks people! If I change the ISO settings on my camera, obviously the gain of the system is increased, amplifying the signal from the sensor. What's not clear to me is where the amplification takes place. I see several possibilities:   In the sensor, by increasing voltage or some other mechanism Via an analog amplifier outside of the sensor Digitally, after the signal has been digitized, but before storing data in the RAW file As a parameter applied solely in creating an image from RAW   If #4 is true, then you could take a 4-stop overexposed RAW picture at ISO 1600, and then in post processing produce a JPEG at ISO 100 that would be the same as if the original photo had been shot at ISO 100.  If #1 or #2 is true, then a RAW file shot at ISO 1600 would actually contain more information about shadows, and an ISO 100 RAW would contain more information about highlights. Is it ok to have mixed grounds and neutrals on bars in a breaker box? So to continue my line of electrical questioning (and perhaps narrow down my ), I took a look through the two panels in this house.  There is a 200amp main service in the basement, which then feeds up to a 100amp sub-panel for the main floor.  This 100amp sub feeds a kitchen (fridge, microwave, dishwasher, gas range), a bathroom, 3 bedrooms, and a living room.  The 200amp main feeds the 100amp sub, 2 bedrooms, a living room, a washer/dryer (gas dryer), a utility room (well pump, pressure tank, gas water heater, gas furnace), and an A/C.  My first concern is that the main service panel has the bare ground and neutral wires mixed on the two bus bars.  Reading around, some say this is OK, other's say it is bad.  Any thoughts?  The sub-panel is wired with grounds and neutrals on separate bars.  My second concern- is a 100amp sub sufficient for the main floor? Easy way to write contents of a Java InputStream to an OutputStream Why does chmod +w not give write permission to other(o) When I run chmod +w filename it doesn't give write permission to other, it just gives write permission to user and group.  After executing this command  chmod +w testfile.txt   running ls -l testfile.txt prints  -rw-rw-r-- 1 ravi ravi 20 Mar 10 18:09 testfile.txt   but in case of +r and +x it works properly.  I don't want to use chmod ugo+w filename. I seen many times how people describe how to's about LBP-2900, and 10-12 steps from start to end of how to download, unpack, build driver, fix driver, make it work... Could be there some .deb package with ready-to-go driver for Ubuntu? This printer is very popular... InputStream returning NullPointerException (Java) Magnetic Field in the presence of a conductor I am studying for my quals and came across an old question that reads like the following:      There are two regions in space separated by an infinite conducting plane. Region 1 has a magnetic dipole of moment $\vec{\mu}$ located a distance $d$ from the plane. What is the magnetic field $\vec{B}$ everywhere in region 1?   I know that for a perfect conductor (superconductor), the normal component of $\vec{B}$ is zero–although I don't exactly know why–but I don't know if that is the case when the problem involves a "conductor" (not explicitly a perfect conductor).  TL;DR: Can anybody give me some insight about the magnetic boundary conditions of a stationary magnetic field in the presence of a conductor?  Thanks! Rigid body simulation not interacting with objects that have an array modifier Is the below a valid circuit implementation and are there any things to watch out for? It's a simplification of the larger circuit but the crux of it is that I need to sink current into two GPIO input pins. The pins are signified by the 1 and 0.  Also, would a diode on the pins be advisable?    Edit: Here's the non-simplified circuit. The question relates to the 3.3v part of it. In essence, the timer and decade counter sequentially pulse current, 'activating' the transistors one at a time. The keypad is a simple matrix keypad. What I'm wondering if I can do is to put a GPIO pin on the line between the keypad and ground. The intended effect is that, when a button on the keypad is pressed, 2 pins will change state - this signifies the row and column.  Note that some of the resistance and capacitance values are notional at the moment. I found this interesting problem in :     Given a point $P$ in space, and given a piece of malleable material of constant density, how should you shape and place the material in order to create the largest possible gravitational field at $P$?   Any ideas? Norm $N$ on $\Bbb Z[\sqrt d]$ is multiplicative, $N(x)$ prime $\Rightarrow x$ irreducible I understand that when I press and release the power button on the top of my iPhone or iPad, it stays on but turns off the screen.  Whereas if I press and hold the power button, a "slide to turn off" control appears on the screen.  Sliding this, as I was told, does turn the unit off, but it does so by suspending the RAM to disk (flash memory, actually.) So, when turning on the iPhone or iPad, it isn't so much rebooting the OS as it is resuming it from flash storage.  How to force the OS to completely reboot on these devices? Word for someone who repeats words Correct interpretation of a "double-negative" in American Pop Culture (i.e. "I ain't afraid of no ghost!") How do I test a nonlinear association? For plot 1, I can test the association between x and y by doing a simple correlation.   For plot 2, where the relationship is nonlinear yet there is a clear relation between x and y, how can I test the association and label its nature? Are user names that represent domains allowed? , are such user names allowed?