I have started the course in commutative algebra. I am facing problem in how to approach this type of questions. How to think of a mapping between these two rings. Help me in this case. here ideal is non-zero.
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ?  I know that a surjective ring endomorphism of a Noetherian ring is also injective, and since there is a natural surjection from $R$ onto $R/I$ we get a surjection from $R$ onto $R$, but the problem is I can not determine the map explicitly and I am not sure about the statement. Please help. Thanks in advance.
Modiagram will not compile Removing OS from Dual Boot computer, and recovering HD space. I have a computer that dual boots Linux Mint, and Ubuntu, I am wondering what the best way would be to remove Linux Mint, then recovering the hard drive space that is resides in so that Ubuntu will be able to access it. UV Unwrapping a Modified Model I modified an existing model by removing unwanted vertices, then selecting surrounding vertices and creating faces one triangle at a time (By pressing 'F').  On UV unwrapping the mesh, the faces which were added by me seem to stretch from the very bottom left of the map, shown below.    How do I create a face so it appears on the UV map as expected? Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left(  \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k}\right \rfloor \right).$$  Please help me with this one.  I tried using the sandwich theorem by first constructing the following  $$ 2 \geq \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k} \right\rfloor \geq -1 $$ and then summing over. But the limit of the two bounds are different so sandwich theorem doesn't help!. Is it possible to construct sharper bounds?  Also I observed that if we ignore question of continuity, etc. then we can write,  $L= \displaystyle \int ^{1}_{0}\left\lfloor \frac{2}{x} \right\rfloor -2\left\lfloor \frac{1}{x} \right\rfloor dx$  Now I am able to compute the above integral. But is the above process correct? If then how can make it rigorous? Or if any other approach is possible, please tell. Prove that there are infinitely many primes and non-primes in the numbers $10^n+1$, where $n$ is a natural number. So numbers are 101, 1001, 10001 etc. Mass dropped on a spring I thought this would be a particularly simple problem but it is turning out to be quite the opposite. I am sure I am doing a very simple mistake.   The problem statement is that there is a mass which is just barely kept on the spring (help by the force equal to the weight of the spring) such that the spring is uncompressed. As soon as I let the object go, it will compress the spring and come to rest at some height. The energy stored in the spring will be equal to the difference in the potential energy at the two height of the spring (compressed and uncompressed),  $$mgh_i -mgh_f= 1/2 kx^2$$ Furthermore, the compression of the spring would just be the difference in the $h_f -h_i$. This gives me two roots for compression,  $$ x=0 $$ And the second would be, $$ mg=1/2kx $$ But then by this, $kx=2mg$ and $kx$ is force by Hooks law, but then does it means that the force compressing the spring is twice the weight of the object? That sounds odd. I was expecting I would just recover Hooks law but I guess I am doing something wrong here. But I am not sure what that is.   Any help will be much appreciated.  P.S. This is not a homework problem. We are designing a project for our school.  Thanks for your time. Installed 14.04 on a non-PAE (Pentium-M), but now I can't update ("This kernel does not support a non-PAE CPU.") Here is what I'm looking at. This isn't my computer. I repair PC's and I had an end user with this low powered laptop running XP so I decided to let him try out lubuntu. The install went smoothly with the non-PAE boot switch, but now I can't update anything. I'd rather not have to format and start over since he's spent time configuring his system. Any tips would be appreciated. I tried sudo apt-get clean, sudo apt-get install -f, sudo apt-get update. No go.  Also his touchpad stopped working, the wireless stopped working, the keyboard works in firefox but not chrome, and in firefox the address bar sometimes turns black and gray for some reason. I was hoping the update would help address some of this stuff.  Here's the message I get trying to update:  Get:1 http://us.archive.ubuntu.com/ubuntu/ trusty-proposed/main linux-image-3.13.0-26-generic i386 3.13.0-26.48 [14.6 MB] Get:2 http://us.archive.ubuntu.com/ubuntu/ trusty-updates/main linux-image-3.13.0-24-generic i386 3.13.0-24.47 [14.6 MB] Fetched 29.2 MB in 2min 19s (208 kB/s)                                          (Reading database ... 169286 files and directories currently installed.) Preparing to unpack .../linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb ... This kernel does not support a non-PAE CPU. dpkg: error processing archive /var/cache/apt/archives/linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb (--unpack):  subprocess new pre-installation script returned error exit status 1 Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.13.0-26-generic /boot/vmlinuz-3.13.0-26-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.13.0-26-generic /boot/vmlinuz-3.13.0-26-generic Preparing to unpack .../linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb ... This kernel does not support a non-PAE CPU. dpkg: error processing archive /var/cache/apt/archives/linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb (--unpack):  subprocess new pre-installation script returned error exit status 1 Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.13.0-24-generic /boot/vmlinuz-3.13.0-24-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.13.0-24-generic /boot/vmlinuz-3.13.0-24-generic Errors were encountered while processing:  /var/cache/apt/archives/linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb  /var/cache/apt/archives/linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) Problem:  If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers  I've come to the conclusion that $a \ne b$ and $a \ne x$ and $b \ne x$ but I guess that won't really help me.. I understand that if the $\gcd$ between two numbers if $1$ then they obviously have no common divisors but where do I go from this point?  Any tips at tackling this would be great. It looks quite easy though I'm still trying to get my hand around these proofs! Any pointers in the right direction would be great.  Thank you in advance, What do all those cryptic number and letter codes in a lens name mean? Adjective for someone who exaggerates emotions to get sympathy What's the word for someone who tries to get sympathy from others by exaggerating their emotions or constantly reminding them of something tragic that happened to them?  I'm looking for an adjective - no disorders or health related definitions please. Is it possible to animate which characters of a text object are assigned to a material? If I want my app to behave differently on a jailbroken iPhone, how would I go about determining this? Use filecontents environment to write a binary file In  of mine, I was trying to show how to embed an audio file into the pdf with the multimedia package. The resulting MWE does not stand on its own because it requires an example audio file.  I guess my first question is, does TeXLive include any example audio files? The second, and potentially less uninteresting question is, can I use the filecontents environment, or some other environment, to create a binary file?  \documentclass{article} \usepackage{graphicx} \usepackage[implicit=false]{hyperref} \usepackage{multimedia}  % What goes in here to produce a usable wav file. Happy to use a different environment. \begin{filecontents}{\jobname.wav} \end{filecontents}  \begin{document} \sound[inlinesound]{\includegraphics{example-image}}{\jobname.wav} \end{document} hard self collision: make particles occupy space Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. List partition labels from the command line Is there a command that will list all partitions along with their labels? sudo fdisk -l and sudo parted -l don't show labels by default.  EDIT: (as per comment below) I'm talking about ext2 labels - those that you can set in gparted upon partitioning.  EDIT2: The intent is to list unmounted partitions (so I know which one to mount). I'm trying to find a story for my father, from the 70s or earlier  Earth is at war with aliens (possibly Martians) and was unable to effectively fight back as the aliens had planetary (or at least city-protecting) force fields through which no bombardment was possible.  Earth eventually wins through invention of a device that allows them to teleport weapons (nukes?) under the force fields.  originally believed to be one of Harry Harrison's 'One step from earth' shorts, but this is not the case.  Thanks in advance for any help here! Is the sizeof(some pointer) always equal to four? I have close to two hundred images where the subject looks like he just walked out of a sauna. It was 36 degrees with humidex, but he was not that red.   For comparison, attaching another picture, same camera, same lens, auto white balance, next day.       So, two questions, 1) how did that happen?, and 2) how do I fix it? I can bring down the red, but then I have to manually bring back the lip color in 200 pictures.   And next time when I deal with an older guy at 36 degrees C I will be shooting indoors.   *Edit: The camera is Olympus EPM 2. I am now guessing that he may have been THAT red, but my eyes tricked me. The last picture is from the beginning of the shoot before he got very hot and had two glasses of wine AND the background worked better for his skin and clothes. I do not have colour management setup, but I will get on it. LaTeX on Stack Overflow? in questions, answers, and comments. Can we get something like this going on Stack Overflow?  I think it’d be appropriate as I at least pretty often want to write something like n^2 and would benefit greatly from prettier markup.