Let $M$ be a finitely generated projective module over a commutative ring $R$ with finitely maximal ideals. Also assume that $M$ has constant rank i.e. rank of $M_P$ as $R_P$-module remains constant as $P$ varies over all prime ideals of $R$.  How to show that $M$ is free ?   I'm allowed to assume that finitely generated projective modules over local rings are free.   My try: Let $J$ be the Jacobson radical of $R$. Then $R/J$ is a finite direct product of fields. Now $M/JM$ is a projective $R/J$-module. Now I'll be done if I can show $M/JM$ is free over $R/J$. Unfortunately I'm unable to show this last part.   Please help.
Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{1}\not\in\mathfrak m_iM_{{\mathfrak m}_i}$ for every $i=1,\dots,n$.    I cannot prove that such an element there exists. I was trying to prove it by induction on $n$. If $n=1$ it is true by Nakayama, so suppose it is true for $n-1$. Then for every $i$ I can find an $x_i\in M$ such that $\frac{x_i}{1}\not\in \mathfrak m_jM_{\mathfrak m_j}$ for every $j\neq i$. If $\frac{x_i}{1}\not\in \mathfrak m_iM_{\mathfrak m_i}$ for some $i$ we are done, so suppose $\frac{x_i}{1}\in\ m_iM_{m_i}$ for every $i$. I don't know how to go on, could you help me?
Why are we limited to six questions per day? Does the Divergence Theorem Work on a Surface? The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by $\partial \Omega$.   My question is that can you use the divergence theorem in $\mathbb{R}^3$ where you're integrating the divergence of a vector field over a surface $S$ in $\mathbb{R}^3$ to get the flux through the edges ($\partial S$) of the surface? So if the surface $S$ is closed in $\mathbb{R}^3$ such as a sphere, then clearly $\partial S = 0$ so the integral of the divergence over this particular $S$ is $0$? Is  purely conceptual or does it represent some real "thing"?  Said another way, is superposition thought to have some tangible physical manifestation or is it simply the lack of physical determinism, except as defined by a wave function?  It seems like my question might be semantic.  For example, it is said that   superposition is obeyed, like a law superposition can be observed, like a phenomena superposition describes behavior   I guess my question is, can superposition be measured like an electrical charge, or is it simply the lack of knowable position. I created a Compute Engine VM (Ubuntu 18.04), and when I log into it using the SSH button in the VM instances list I'm logged in as user username.  After a while, I realised I needed to be able to use another SSH client as well. As per  I enabled OS Login by adding a key-value pair in metadata, where key is enable-oslogin and value is TRUE.  I then tried to add my existing SSH key:  gcloud compute os-login ssh-keys add --key-file path/to/key.pub --ttl 0  which was successful except one major problem: it has added the key for user username_gmail_com and will only allow me to ssh to my VM as that user:  ssh username_gmail_com@myinstance  And it has also created that user on my VM. From the web interface, I'm logged in to the VM with that new user as well (/home/username_gmail_com). Of course, I can still access the original user's data (/home/username). When I disable OS Login the web interface logs me in as username again and SSH from another client is disabled.   Why is the username different when enabling OS Login? Can I change the username with OS Login enabled?   Edit  There has been an addition to the docs explaining exactly this question. Check here:  and on  all the way at the bottom under "Expected login behaviors". How specific does "beast you have seen" have to be for Wild Shape? Starting at 2nd level, you can use your action to magically assume the   shape of a beast that you have seen before.   This doesn't say anything about the gender or age of the beast, so can a druid change into a female bear if he has only seen male bears? Can he change into a bear cub if he has only seen adult bears? What about hens/roosters/chickens, which look quite a bit different? Or does the gender of the animal shape just match the gender of druid? I have filed my 2012 tax return jointly with my wife, along with my wife's ITIN application, back in late March.  In April, I got a letter for IRS ITIN department saying that we should send my wife's original passport to them. So we sent my wife's passport on May 15th. USPS tracking number confirmed that IRS received it on May 20th.  However, last week (end of June) we have received another letter from IRS ITIN department saying that our ITIN application has been rejected, rejected because we have not sent the original identification document that they requested!  Okay, let's put our tax return and ITIN aside. Where is my wife's passport? Right now my wife does not have her personal identification, nor proof of legal status in U.S. Could IRS have lost it?  I tried to call the IRS several times. One time I waited for 1 hour and 20 minutes to talk to a person, and that person said he has only 10 minutes to talk before he has to leave. He said they'll investigate and someone(?) will call us back soon, but never called back. I wrote them a letter, but no response. I tried calling several times more, but machine is saying 'sorry too many calls right now, we can't accept your call'.  What can I do? fontenc changes sans serif bold font in KOMA-script Consider the following MWE:  \documentclass{article} %\usepackage[T1]{fontenc}  \begin{document}  \textbf{\textsf{Some text}}  \end{document}   The output of the above code is     and doesn't change if I uncomment the line  %\usepackage[T1]{fontenc}   Consider now the following MWE with scrartcl class instead of article:  \documentclass{scrartcl} %\usepackage[T1]{fontenc}  \begin{document}  \section{A section} \textsf{Some text} \end{document}      Uncommenting the line  %\usepackage[T1]{fontenc}   you get    As you can see, the sans serif font of the title is less  bold than without fontenc, while the normal text is not changed.  Does fontenc, combined with a KOMA-script class, load a different sans serif font, but only for bold face? Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? How to return only the Date from a SQL Server DateTime datatype Requested value 'CheckBoxes' was not found There's a weird thing happening with my Windows application. I have created a Sharepoint utility to automate the creation of content-types and site-columns. At a point where I try to get my available fields, I get this error:     An unhandled exception of type 'Microsoft.SharePoint.Client.ServerException' occurred in Microsoft.SharePoint.Client.Runtime.dll   Additional information: Requested value 'CheckBoxes' was not found.   It is throwing an exception at:  ClientContext clientContxt = new ClientContext(clientContext.Url);         clientContext.Credentials = clientContext.Credentials;         Web web = clientContext.Web;          clientContext.Load(web, w =&gt; w.AvailableFields);         clientContext.ExecuteQuery(); //**exception thrown here**   Kindly please suggest what is wrong here? Difference Between Sequential Workflow & State machine workflow? What is the difference between Sequential Workflow and Stater machine workflow? What context should I take account to when choosing one of them? What's the difference between a method and a function? I wish to show that $\mathbb{Q}(\sqrt2, \sqrt3)$ is not isomorphic to $\mathbb{Q}(\sqrt[4]2)$.    What I did so far was to show that $\mathbb{Q}(\sqrt2,\sqrt3) =\text{span}\{1,\sqrt2,\sqrt3,\sqrt6\}$.  Assuming they are isomorphic, denoting $f$ the isomorphism, then $f(1) = 1$, so $x^2-3$ has a root in $\mathbb{Q}(\sqrt2,\sqrt3)$ so $\sqrt 3\in\mathbb{Q}(\sqrt[4]2)$. ($\sqrt2 \in \mathbb{Q}(\sqrt[4]2)$, $\sqrt[4]2\cdot\sqrt[4]2=\sqrt2)$.   Thus under the assumption, $\mathbb{Q}(\sqrt2, \sqrt3)\subseteq \mathbb{Q}(\sqrt[4]2)$. As a vector spaces of the same dimension over $\mathbb{Q}$, it means that $\mathbb{Q}(\sqrt2, \sqrt3)= \mathbb{Q}(\sqrt[4]2)$.  Now all is left to do it to show a contradiction: $\sqrt[4]2\in\mathbb{Q}(\sqrt2,\sqrt3)$, then $\sqrt[4]2 = a+b\sqrt2+c\sqrt3+d\sqrt6$ so $2=(a+b\sqrt2+c\sqrt3+d\sqrt6)^4$, I believe that opening the expression, I would find out that it can't be an element of $\mathbb{Z}$. However this is a really inelegant way (if it really works), and I wish for a better way to do so.  An other approach: $\mathbb{Q}(\sqrt2,\sqrt3) = \mathbb{Q}(\sqrt2 + \sqrt3)$ by the irreducibility of $x^4-10x^2+1$ over $\mathbb{Q}$ (an some dimension theorem usage). Also, $x^4-10x^2+1$ splits in $\mathbb{Q(\sqrt2, \sqrt3)}$.   Is there a way to show that this polynomial doesn't split over $\mathbb{Q}(\sqrt[2]4)$? with different method then assuming that $\sqrt3\in\mathbb{Q}(\sqrt[4]2)$ and getting a contradiction? Maybe the fact that $\sqrt[4]2$ has complex roots might help? Right now  is fairly broken -- it's a simple descending sort by views. As Joel said , it is &quot;a self-fulfilling prophecy.&quot; But this is not intentional, it's only because we haven't had time to improve it yet! As I sit down to write a better algorithm, I thought you guys might have some insight into what would make a good formula to determine the &quot;hot&quot; questions -- over the last 24 hours, the last week, the last month, and all time. Here are the variables we have to work with:  Votes [aka Score] Views Answers Answer Votes [aka Score] Whether the question has an accepted answer or not Time question was originally asked Time of last activity on question Reputation of asker Reputation(s) of answerers  Joel cited the reddit formula as an example, but I think our site is a little different than just a basic link submission site. Plus, we have more variables and data to work with. I found a  describing a few formulas: Y Combinator Hacker News  (p - 1) / (t + 2)^1.5 p = votes (points) from users t = time since submission in hours  Reddit  t = (time of entry post) - (Dec 8, 2005) x = upvotes - downvotes y = {1 if x &gt; 0, 0 if x = 0, -1 if x &lt; 0) z = {1 if x &lt; 0, otherwise x} log(z) + (y * t)/45000  Thoughts? Ideas? I don't want to overthink this; I just want the &quot;hot&quot; questions to be relevant and useful to us. So the simpler the algorithm the better! Based on my analysis of the above and the comments so far, here's the second version of what I have implemented so far. This might suck. I don't know:  (log(Qviews)*4) + ((Qanswers * Qscore)/5) + sum(Ascores) -------------------------------------------------------- ((QageInHours+1) - ((QageInHours - Qupdated)/2)) ^ 1.5  Note that accepted answers weight not at all in hotness. This is intentional, as I feel accepted answers are a fine social contract, but not a good data point for question or answer quality. This algorithm will heavily favor questions with LOTS of answers, as the sum(Ascores) are now included -- one assumes if there are lots of answers, there will be a lot more voting on the answers, too. Update: Note that this formula is what is used on the  linked from the homepage of each site. It is not the formula used to determine the  list. See . Kinematic Viscosity How would you define ? What does it physically represent? Around the Internet I've found it defined as just a ratio, and that's it.  I saw in an answer that I can think of it as "diffusivity of momentum". What does this mean? Getting around the FAT32 4GB file size limit Can you raise a number to an irrational exponent? The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a number to an irrational number? Obviously it is not so simple to break it down like above. Does an irrational exponent still have a well formed meaning?  The only example that comes to mind is Euler's identity, but that seems likes a pretty exceptional case. What about in general? Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. Is there any way of making a snow golem survive in hot biomes? Snow golems won't survive in the Nether (neither spawned in nor built ones). Is there a way to make a snow golem survive in the Nether and other hot biomes in vanilla Minecraft (i.e., without using mods)? High p-values for logistic regression variable that perfectly separates? I'm using R to run some logistic regression. My variables were continuous, but I used cut to bucket the data. Some particular buckets for these variables always result in dependent variable being equal to 1. As expcted, the coefficient estimate for this bucket is very high, but the p-value is also high. There are about ~90 observations in either these buckets, and around 800 total observations, so I don't think it's a problem of sample size. Also, this variable should not be related to other variables, which would naturally reduce their p-values.  Are there any other plausible explanations for the high p-value?  Example:  myData &lt;- read.csv("application.csv", header = TRUE) myData$FICO &lt;- cut(myData$FICO, c(0, 660, 680, 700, 720, 740, 780, Inf), right = FALSE) myData$CLTV &lt;- cut(myData$CLTV, c(0, 70, 80, 90, 95, 100, 125, Inf), right = FALSE) fit &lt;- glm(Denied ~ CLTV + FICO, data = myData, family=binomial())   Results are something like this:  Deviance Residuals:       Min        1Q    Median        3Q       Max   -1.53831  -0.77944  -0.62487   0.00027   2.09771    Coefficients:                 Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)     -1.33630    0.23250  -5.747 9.06e-09 *** CLTV(70,80]     -0.54961    0.34864  -1.576 0.114930     CLTV(80,90]     -0.51413    0.31230  -1.646 0.099715 .   CLTV(90,95]     -0.74648    0.37221  -2.006 0.044904 *   CLTV(95,100]     0.38370    0.37709   1.018 0.308906     CLTV(100,125]   -0.01554    0.25187  -0.062 0.950792     CLTV(125,Inf]   18.49557  443.55550   0.042 0.966739     FICO[0,660)     19.64884 3956.18034   0.005 0.996037     FICO[660,680)    1.77008    0.47653   3.715 0.000204 *** FICO[680,700)    0.98575    0.30859   3.194 0.001402 **  FICO[700,720)    1.31767    0.27166   4.850 1.23e-06 *** FICO[720,740)    0.62720    0.29819   2.103 0.035434 *   FICO[740,780)    0.31605    0.23369   1.352 0.176236     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for binomial family taken to be 1)      Null deviance: 1037.43  on 810  degrees of freedom Residual deviance:  803.88  on 798  degrees of freedom AIC: 829.88  Number of Fisher Scoring iterations: 16   FICO in the range [0, 660) and CLTV in the range (125, Inf] indeed always results in Denial = 1, so their coefficients are very large, but why are they also "insignificant"?