This is easy to see in the ring of integers. In fact, the ideals don't even have to be prime. It's enough to be coprime. Then their GCD is 1, so 1 can be written as a linear combination of the generators of your ideals, and once you get 1 in the ring, you have the whole ring.  Is this true for rings in general? If not, what conditions must we impose on the ring (euclidean domain, principal ideal domain, etc.) in order to make it true?
Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when is it the case?) that $\mathfrak{p}_1 + \mathfrak{p}_2 = K$ ?  I ask because I'm working through some lecture notes which frequently take some $K$ complete with respect to a discrete valuation, a prime ideal in a valuation ring  $\mathfrak{p} \subset \mathcal{O}_K$, a finite extension $L/K$, and then use the fact $\mathfrak{p}\mathcal{O}_L$ factors uniquely as some $\mathcal{P}_1^{a_1} \ldots \mathcal{P}_r^{a_r}$ (where $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$). Then we say $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$ $(\forall$ $ i \neq j)$.   However it isn't clear to me, because we've made so many different assumptions along the way, what the reason is that $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$. Is it because these primes occur in the factorisation of $\mathfrak{p}$, is it because of some property of Dedekind domains or fields, or is it not really possible to say out of context? Many thanks, and if anything needs clarification please ask.
A Pokemon which has the Technitian ability gets x1.5 boosted attack power for all moves used which have an attack power of 60 or less (so a move with 60 attack power will actually have 90 attack power). STAB (Same type attack bonus) gives a 50% boost to damage, assuming the attack has the same type as the Pokemon using it (so a Grass Pokemon using Razor Leaf will do 1.5 times the damage). However, I have seen contradicting information, so currently I do not know whether STAB boosts the actual damage or the move's attack power.  What I would like to know, first, is the precise manner in which a STAB is applied to an attack. Secondly, I would like to know what happens if an attack which would be affected by the Technician ability is also affected by STAB (though this answer might vary depending on the actual way that STABs work).  For example, my Scyther has the Technitian ability and attacks using the "Wing Attack" attack, which is a Flying-type attack with 60 attack power. Since Scyther is a Bug/Flying type pokemon he should also get a STAB for this attack, so just what is the eventual attack power/damage in this case?  (Please answer for 4th generation mechanics) Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains?  I heard many reasons, but I'm not sure which is true.   Some say it's just a matter of beauty, so Lagrangians are more beautiful because they don't break/separate the space-time variables (so space-time is a single variable, like in the Klein-Gordon Lagrangian and Hamiltonian).  Some say that Hamiltonians are not always Lorentz invariant.  Could someone explain in a little bit more details? Should Dalvik Cache be wiped in Android OS that supports ART? When switching custom ROMs (in ART), is it necessary to wipe the Dalvik Cache?  Wiping the Cache is necessary, but should wiping Dalvik (which doesn't exist in Android Lollipop and afterward) be necessary?  What impact will it cause if we wipe Dalvik (something not present)?  In ART, what does the Dalvik cache contains? (is it an empty folder?) PHP regex delimiter, what's the point? $f(x)=x^3$ is convex nor concave without to derivate QThread and QTimer I have two partitions, one of them has a file system and the other is unallocated.  How to merge two partitions? text processing rows to columns for a block Normal subgroups of $S_4$ Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?  In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups? Checkbox in listview control Can you have a multicolumn listview control where one of the columns is a checkbox?  Example code or links would be greatly appreciated.  I am using visual studio 2005 In my case I'm talking about Assassin's Stance, but I'd like to know the answer for stances in general.  I feel like I might be missing something incredibly obvious caused by missing a word or two (or a paragraph), but I can't find anything that explicitly states whether or not I can use the Tome of Battle stances with ranged weapons like the bow (or crossbow).  I realize that half the classes have "blade" in their names so using ranged weapons is not in the spirit of things, but I'm looking for an explicit answer (or as close to one as possible) from any WotC official resource. If this has already been answered I'm sorry, but I could not find the answer myself from searching. I finally got fed up when wanting to read about bash's read and it's -s option with man bash. I found the right spot eventually (around line 4500), but it was a frustrating as usual, since both /read and even /\s-s\s searches have way too many matches.  So, the question is: How can I read long man pages efficiently, or get same information in other ways, locally? As a specific example, how to reach the relevant documentation after seeing read -s pwd in a shell script? A good answer could be a shell script snippet, or hint about some tool and how it is used, or something else entirely, as long as it helps in finding the right spot to read.  Note: I'm not tagging with  because I want the question to be about man page reading in general, even though that quite possibly is the most commonly encountered humongous man page. Can't install rtl8192eu-linux-driver on UBUNTU 18.10 Proof of Funds for Canadian Visa I am a Romanian citizen.  Our co-national friends who are established in Canada will provide me and another person free accommodation and food. We are both EU citizens and we will stay in Canada 1 month.  These days we are gathering documents for our online visa application and we need to prove that we have enough money to spend there.   How much money is generally required in order to be sure we are eligible? We don't plan to spend more than 1000 EUR, but I intend to bring at least 2000 CAD. Is it enough? ROC vs Precision-recall curves on imbalanced dataset I just finished reading  discussion. They argue that PR AUC is better than ROC AUC on imbalanced dataset.   For example, we have 10 samples in test dataset. 9 samples are positive and 1 is negative. We have a terrible model which predicts everything positive. Thus, we will have a metric that TP = 9, FP = 1, TN = 0, FN = 0.  Then, Precision = 0.9, Recall = 1.0. The precision and recall are both very high, but we have a poor classifier.  On the other hand, TPR = TP/(TP+FN) = 1.0, FPR = FP/(FP+TN) = 1.0. Because the FPR is very high, we can identify that this is not a good classifier.  Clearly, ROC is better than PR on imbalanced datasets. Can somebody explain why PR is better? Why is polyploidy lethal for some organisms while for others is not? is the multiplication of number of chromosomal sets from 2n to 3n (triploidy), 4n (tetraploidy) and so on. It is quite common in plants, for example many crops like wheat or Brassica forms. It seems to be rarer in animals but still it is present among some amphibian species like Xenopus.  As I know in mammals polyploidy is lethal (I don't mean tissue - limited polyploidy). I understand that triploidy is harmful due to stronger influence of maternal or paternal epigenetic traits that cause abnormal development of placenta, but why there is no tetraploid mammals? Defining macros with arguments With newcommand I can define customized command like \ABC. Is there any way to define a command with arguments. For example I want \brat{A}{B} be the same as [\![A,B]\!]. How to disable SSLv2 or SSLv3? How do I get current date/time on the Windows command line in a suitable format for usage in a file/folder name? This question is here so we can offer users who are looking for information on how to make one or more common or basic redirects in Apache using the htaccess file. All future questions pertaining to finding information that is that is covered by the question should be closed as a duplication of this question. .  Whats the point in this question? The idea while not perfect is catch the most commonly asked questions regarding redirects using the  htaccess on the Apache platform either on  some type of lamp or a live server. The type of answers should be generally those that you could imagine are used by 100,000’s of sites world-wide and are constantly asked here at Pro Webmasters repeatedly over and over in various forms.  A few examples of the type of answers we are looking for:   How can I redirect non-www to www? How can I redirect a sub domain to the main domain? How can I redirect a sub folder from domain to a root or a subdomain?  How can I redirect an old URL to a new URL?   A few examples of the types of answers that we are not looking for:   Answers that do not involve a redirect. Any answers relating to NGinx, IIS or any other non-Apache platform. Answers that involve custom and complex string or query removals.   Resources for Advanced to Complex Mod_Rewrite Rules:      Please note: that this question is still in construction and may need some refining either by myself or a real moderator of Pro Webmasters, if you have any concerns or questions .