I asked this question before but did not get an answer, I tried solving it myself and I think I'm heading somewhere, I just need a push in the right direction.  Let $\phi: R \to S$ be a homomorphism of commutative rings. Suppose that $\phi$ is surjective and that $R$ is a principal ideal ring (Meaning all the ideals are principal ideals, spanned only by 1 term).  Show that $S$ is also a principal ideal ring.  My "psuedo" solution  We know that $R/ker(\phi)$ is isomorphic to $Im(S) = S$ ($\phi$ is surjective).  We also know that $ker(\phi)$ is an ideal, and we know that if $R$ is a principal ideal ring and $I$ is a prime ideal then $R/I$ is also a principal ideal ring.  So I wanted to show that $ker(\phi)$ is a prime ideal, that would solve the question. But it doesn't have to be...Let's look at $ab \in ker(\phi)$:  $ab \in ker(\phi)$ implies $\phi(ab) = 0$ which implies $\phi(a)\phi(b)=0$ which implies $a \in ker(\phi)$ or $b \in ker(\phi)$ IF AND ONLY IF $S$ has no zero divisors. We we're not given that. it could be possible to find $\phi(a),\phi(b) \in S$ such that their product is zero and neither of them is zero.  And that is where I am stuck.
I was working on a worksheet for class and came across this problem:      Let $\varphi:R\to S$ be an onto ring homomorphism with the property that every ideal of $R$ is principal. Show that the same must be true for $S$.    I honestly do not know how to do this one!
Is the tag info page not available on mobile theme? I was surfing SO on my iPhone, and wanted to see the info (tag-wiki) page for a tag, but didn't find it. Clicking on the tag always leads to the list of question, and there is no other link to the tag wiki. Is it not available on the mobile site? I am unable to understand what exactly is critical value.My question here is why do we fail to reject a null hypothesis when the test statistic is within the range of critical value.  What i am trying to ask is what is the property of the critical value that makes the hypothesized value to fall within the range implied by the critical value.  Thanks in advance Allow us to vote to delete from the Android app Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: When I'm not exploring fresh caves or lands in minecraft, I'm busy trying to find deposits of diamond, gold and iron deep in the earth. So far I've just been digging around just above bedrock on a whim to look for these rare minerals but I've started thinking that I should be taking a more controlled, efficient approach to mining.  What is the optimal mining strategy that yields the most rare minerals while minimizing blocks removed and time spent digging and traveling?  Edit: The existing answers are great for mining, but no one has addressed travel time to and from the mine which is usually ~60 blocks downward and incredibly tedious. Solutions? Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain. The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable I'm trying to prove this conclusion but have some problems with one of the steps.  Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.  We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated convergent theorem, we have for any $t$  $$ \lim_{n\rightarrow\infty}e^{it\mu_n-t^2\sigma_n^2/2}=\lim_{n\rightarrow \infty}\phi_{X_n}(t) = \lim_{n\rightarrow \infty}\mathbb{E}\left[e^{itX_n}\right] = \mathbb{E}\left[e^{itX}\right] = \phi_X(t) $$  this is the step that I cannot figure out: $e^{it\mu_n-t^2\sigma_n^2/2}$ converges for any $t$ if and only if $\mu_n$ and $\sigma_n$ converges.   Let $\mu=\lim_n \mu_n$, and $\sigma=\lim_n\sigma_n$, then $\phi_X(t)=e^{it\mu-t^2\sigma^2/2}$, which proves that $X$ is a Gaussian random variable.  Why can we get that $\mu_n$ and $\sigma_n$ converge? This looks intuitive for me, but I cannot make a rigorous prove. The screen of my phone is physically broken, so it doesn't display anything, but the touch aspect does seem to be working, as I can unlock my phone, and get haptic feedback.  Before I go to get it replaced, I'd like to back up some personal data from the built-in memory to the SD card. I'm fairly certain that if I can get adb to recognize the device, then I can use  to see where I am touching the screen and do what I need to do. Problem is, when I plug it in, it's not recognized, while my emulator is.  I suspect that I might not have USB debugging enabled, which I think is necessary for adb to work. If this is the case, can someone give me screenshots of their Droid X (running 2.3.X) going from the home screen, through all the menus, to activating debug mode? Hopefully I can do it by dead reckoning and adb from there.  Edit: I haven't changed anything about the ROM or launcher; it's all stock. B2B application: users has its own users B2B Scenario I am trying is that:  The following Roles exists in the system:  Super Admin Super Admin Staff Owner (client) Admin (owner i.e. client and creates admin to define sales counters) Sales Counters  That means there are users who'll create their own sub users e.g.  [FLAG-A] ROLE &quot;SUPER ADMIN&quot; creates a ROLE &quot;Owner&quot; and &quot;OWNER&quot; creates its own sub users ROLES i.e. &quot;ADMN&quot; &amp; &quot;SALES COUNTER&quot;. Each role obviously has its info to login to the system I tried to design database as following:  In this diagram I am assuming that main user creates sub user that in my view many to many relationship so I have to add table between them as &quot;USER_PERSON&quot;. Problem is to set &quot;WHO CREATED USER&quot; in the system. I mean which user has created sub user as described in [FLAG-A]  Please check this diagram How to prevent iMessage syncing without access to device I foolishly logged into someone's iPad with my Apple ID. This person is now receiving my full communications on iMessage. I know I can remove my account from that iPad but I don't have access to it. How can I stop the device from getting my iMessages without physical access? Making layer or group private/hidden/invisible in Layers Panel of QGIS I have a number of layers in the Layers Panel inside a group but I don't want them shown (there's quite a lot so it looks untidy). These layers are also joined to several others. Is it possible to make these layers invisible to the user (i.e. not shown in the Layers Panel, it is not about setting the visibility of the layer) but can still be joined to other layers, or perhaps to make the group invisible? Closest I found was this post but hoping there's a more direct approach: Modern electronics industry has developed very efficient LEDs with nice and fancy colors (“true green” , “cyan”, and “blue”) based in InGaN technology, or similar. Some of these LEDs are specified as having forward voltages of 3.2 to 4.2 V. Here are examples of such LEDs, , or surface-mount .  But the usual MCUs are powered from 3.3 V supply, and the GPIO output is usually at 3.1-3.2 V, which seems to be barely enough to drive these LEDs . How can I use these LEDs as indicators of some signal statuses, or power rail status? Do I need a series resistor R1 here?    If yes, how do I choose its value?  DISCLAIMER: This "question-answer" comes as elaboration of the original question , to address the particular use case of LEDs as logic state indicators. I need to use an alias in the WHERE clause, but It keeps telling me that its an unknown column. Is there any way to get around this issue? I need to select records that have a rating higher than x. Rating is calculated as the following alias:  sum(reviews.rev_rating)/count(reviews.rev_id) as avg_rating Why does the Cauchy distribution have no mean? From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean? I have been thinking a bit about Morita equivalence   and I would like to know whether it also applies to subrings such as right or left ideals.  And, if so, how specifically? Could you illustrate if with the example of 2x2 matrices with zero entries in the bottom row? Since Morita equivalence concerns non-commutative rings, I am assuming it should apply, but I am just guessing.  Thanks in advance. Search only for the currently active language Replaced wrong car in garage in GTA Online I was asked to replace a car in order to put another car in my garage. However I thought I would be able to swap the car I wanted in and take out the other car that was already in there so I could bring it to my other garage. Can I get my car back? Javascript file uploads My 2nd great-grandmother, Margaret Elizabeth Armegust or Arbingast, was born 22 Sept 1837 in Ohio and died 24 July 1929 at the Richmond State Hospital, Wayne, Indiana, United States.  She lived for a time in Sparta, Noble, Indiana and Stock, Harrison, Ohio. She was married to John B. Miller on 22 Oct 1859 in Noble and they had at least five children: James, Samuel, Elias, Julia, and John.  I am searching for records from the State Hospital to learn of her diagnosis and treatment. Batch jobs error emails report:      The transaction log for database 'WebDB' is full.   No, it isn't. Its 90% free. The transaction log backups run normally. My hardware guy tells me the hardware is running perfectly, with plenty of space. SQL Server 2008 SP3, but not R2.  I'm a developer, we no longer have a DBA. So it falls to me to try and fix this. While finding escape velocity, does human mass counts as the mass of Earth? Escape velocity depends upon the mass of earth. my doubt is whether human mass adds into it or not? If it add then Earth's escape velocity would be increasing!