The set $A$ of all lower triangular $2\times 2$ matrices with entries in $\mathbb Z$ is a subring of $M_2(\mathbb Z)$. However, what are its ideals? I couldn't find them.   Suppose $I$ is an ideal of $A$. The elements of $I$, being also elements of $A$ have to be lower triangular and $I$ should be closed under left and right multiplication by lower triangular matrices. These facts couldn't lead me anywhere.  Edit: I saw  and the answer there went far above my head. I have no idea what a module is. Please help me do this without resorting to all of those advanced technique
Suppose $T$ is the subset of $M_2(\mathbb{Z})$ of lower triangular matrices, those of form $\begin{pmatrix} a &amp; 0 \\ b &amp; c\end{pmatrix}$. So $T$ is a subring. Now I know that the ideals of $M_2(\mathbb{Z})$ are all of the form $M_2(I)$ for $I$ and ideal of $\mathbb{Z}$.   However, is there a nice way to describe all the ideals in $T$ specifically, or does it not behave quite as nicely?
I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? Classes and lessons at Hogwarts are presumably scheduled at fixed times of day. The first class might start, for example at nine in the morning.  To be on time for one's lessons, it is therefore crucial to know the current time but we know that electric watches of course do not work at Hogwarts because of all the magical interference. Are there magic watches? Is there maybe a spell to tell the time?  How do people at Hogwarts usually tell the time? What is the origin of elements heavier than iron? In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. What happen if I install VirtualBox Guest Additions on a host? I have a bootable USB drive with Ubuntu installed.   This morning I Installed VirtualBox and booted my Ubuntu drive in a VM. I then installed the guest additions in Ubuntu.   My question/curiosity is, what would happen if I boot the drive again outside of the VirtualBox VM, using a physical computer?  Will my Ubuntu work normally or is something going to change? Find all $x$ such that $x^6 = (x+1)^6$. Can anyone point me to a reference (preferably either something online or something a small liberal arts school would be likely to have in its library) that goes through a derivation of the minimum uncertainty wavefunction in more detail than in the Griffiths?  Edit I've moved the second part of my original question to a separate post: What use are Steam Trading Cards? Is it purely a 'gotta-have-them-all' mindset, or do they actually serve some purpose?  I notice that you are able to sell them on a community market. For a test I listed one, and someone bought it for 50p. Not exactly a huge amount, but I don't understand why someone would spend real money on them.  Am I missing something?   Are they redeemable for games or money off vouchers or something 'tangible' that I'm missing? Simple question, but one that I've been curious about...is there a functional difference between the following two commands?  String::class String.class   They both do what I expect -- that is to say they return Class -- but what is the difference between using the :: and the .?  I notice that on those classes that have constants defined, IRB's auto-completion will return the constants as available options when you press tab after :: but not after ., but I don't know what the reason for this is... Kinetic energy conservation in a collision Kinetic energy conservation in collision.  In any type of collision type of collision if we take the system to comprised of both bodies then the net external force is zero. So the work done by external force is also zero. By the relation $W=K_f-K_i$ as $W$ is zero hence kinetic energy is conserved. But this not the case.   Can you tell me what is the flaw in my argument? Some people say that the argument has neglected the fact that some kinetic energy is transformed into heat. But heat is created due to the interactions between bodies during collision which is thus an internal force and thus shouldn't affect kinetic energy. Conditionally enabling systemd files through Debian packaging I've created a  package which installs a service.  On our embedded devices, I want this package to automatically enable the service.  On our developer workstations, I want the developers to systemctl start foo manually (it's a heavy service, and so it just consumes resources if run all of the time on a desktop environment).  How can I go about prompting the user for their decision during the apt-get step? Is that the best solution?  Note, I've created the package using dh_make and debhelper and enabled it with:  %:     dh $@ --with=systemd  override_dh_systemd_enable:     dh_systemd_enable --name=foo foo.service Apply subsurf modifier to only part of an object How can I apply a modifier only to a part of an object? I want to model a goblet with a pedestal, and if I add a subsurf, the pedestal, of course, gets an unwanted round shape. How can I avoid that? I have tried Googling it, but, to my surprise, found nothing. I use a Asus laptop with dual-boot Ubuntu and Windows 8.1  Wifi is on but I am not able to detect Wifi networks.  When I run :  lspci -knn | grep Net -A3; rfkill list   This is what happens:    I get  acer-wireless: Wireless LAN     Soft blocked: yes setTimeout and "this" in JavaScript I have a method that uses the  function and makes a call to another method. On initial load method 2 works fine. However, after the timeout, I get an error that says method2 is undefined. What am I doing wrong here?  ex:  test.prototype.method = function() {     //method2 returns image based on the id passed     this.method2('useSomeElement').src = "http://www.some.url";     timeDelay = window.setTimeout(this.method, 5000); };  test.prototype.method2 = function(name) {     for (var i = 0; i &lt; document.images.length; i++) {         if (document.images[i].id.indexOf(name) &gt; 1) {             return document.images[i];         }     } }; Grammatical complements for "allow" Are any of these verb phrases grammatical?        allows the user of modeling and resolving     allows the user to model and resolve      Which version of the following sentence is correct/better?        In particular, it allows to acquire a signal using a sample rate significantly lower than the one dictated by the Nyquist criterion.   In particular, it allows acquiring a signal using a sample rate   significantly lower than the one dictated by the Nyquist criterion.      I searched Google Scholar for "" and "", and it seems people use both more or less with the same frequency. C# List Sort by x then y Similar to , we want to sort by one element, then another.  we want to achieve the functional equivalent of   SELECT * from Table ORDER BY x, y     We have a class that contains a number of sorting functions, and we have no issues sorting by one element. For example:    public class MyClass {     public int x;     public int y; }    List&lt;MyClass&gt; MyList;  public void SortList() {     MyList.Sort( MySortingFunction ); }   And we have the following in the list:    Unsorted     Sorted(x)     Desired ---------    ---------    --------- ID   x  y    ID   x  y    ID   x  y [0]  0  1    [2]  0  2    [0]  0  1 [1]  1  1    [0]  0  1    [2]  0  2 [2]  0  2    [1]  1  1    [1]  1  1 [3]  1  2    [3]  1  2    [3]  1  2   Stable sort would be preferable, but not required. Solution that works for .Net 2.0 is welcome. What is the isomorphism between the fields $(Z_2[x]^{<3},+_{x^3+x^2+1},\times_{x^3+x^2+1})$ and $(Z_2[x]^{<3},+_{x^3+x+1},\times_{x^3+x+1})$? Plot $|z - i| + |z + i| = 16$ on the complex plane Plot $|z - i| + |z + i| = 16$ on the complex plane   Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will form an ellipse. I was having trouble getting the equation of the ellipse algebraically.  I get to the point:  $x^2 + (y - 1) ^2 + 2\sqrt{x^2 + (y - 1)^2}\sqrt{x^2 + (y + 1)^2} + x^2 + (y+ 1)^2 = 256$  $2x^2 + 2y^2 + 2\sqrt{x^2 + (y - 1)^2}\sqrt{x^2 + (y + 1)^2} = 254$  It seems like I'm and doing something the hard way. I wanted a list of sub-domains of Stack Exchange that are not mentioned over here   Some of the sub-domains I found which were not mentioned are       Are there any other which are not mentioned?    Mentions by , ,             Other mentions below which are not mentioned in "All Sites" but are mentioned in the drop down menu on the top left of the Stack Exchange website         Any other website that's left out ? Start codon AUG also codes for methionine and without start codon translation does not happen. And even the ambiguous codon GUG codes for methionine when it is first. So does this mean that all proteins start with methionine as the first amino acid. What do you call exaggerations like "I'm starving"? For example, when you are a little hungry and you say "I'm starving", or when you are so tired and you say "I'm dying". What do you call these type of expressions? Just exaggerations? I don't know how to explain this better.