Let $R$ be a local ring with maximal ideal $I$ and residue field $k=R/I$ and $M$ finitely generated $R$-module.   My question is why $$M \otimes_R k\cong M/IM,$$ where $M \otimes_R k$ and $M/IM$ are $R/I$-modules.
Possible Duplicate:          In  the following claim was mentioned:  $R/I \otimes_R M \cong M / IM$  So I tried to prove it. Can you help me finish my proof? Thanks!    We recall that $M \otimes_R -$ is a covariant right-exact functor that it is exact if $M$ is flat and we observe that the following is an exact sequence: $$ 0 \to IM \xrightarrow{i} M \xrightarrow{\pi} M / IM \to 0$$  $R/I$ is an $R$ module since $R/I$ is a subring of $R$ closed with respect to multiplication from $R$ but it is not necessarily flat hence we only get exactness on one side:  $$  (R/I) \otimes_R IM \xrightarrow{id \otimes i} (R/I) \otimes_R M \xrightarrow{id \otimes \pi} (R/I) \otimes_R M / IM \to 0$$  Then $$ \mathrm{Im(id \otimes \pi)} = (R/I) \otimes_R M / IM  \cong (R/I) \otimes_R M / \mathrm{Ker} (id \otimes i) $$  so we want to show two things:   (i) that $\mathrm{Ker} (id \otimes i) = \{0\}$. To this end let $r + I \otimes im \in (R/I) \otimes_R  IM $ and assume $ id \otimes i(r + I \otimes im ) = r + I \otimes im = 0 + I \otimes 0$. I'm not sure how to proceed from here.   (ii) and that $M /IM \cong (R/I) \otimes_R M / IM $  Can you help me? Thanks.
NOTE: This is a Master Canonical Post for the issues observed on February 25, 2020.  All issues on Feb 25 relating to Python updates not found should be dupe-linked here.  Errors have been seen such as this today, February 25, 2021: Failed to fetch http://security.ubuntu.com/ubuntu/pool/main/p/python2.7/libpython2.7-minimal_2.7.17-1~18.04ubuntu1.3_amd64.deb 404 Not Found  Is there something up with the repositories? Finding an Inverse Function and Composition of Functions? Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ bash profile works for user but not sudo Is a comma necessary in "What’s funny, is …"? In the sentence     What’s funny is Cat’s dry humour.   Wouldn’t it be better to put a comma after “funny”:     What’s funny, is Cat’s dry humour.   This question entails a (at the time of writing) heated discussion in the . Why doesn't backpropagation work when you initialize the weights the same value? Why doesn't backpropagation work when you initialize all the weight the same value (say 0.5), but works fine when given random numbers?   Shouldn't the algorithm calculate the error and work from there, despite the fact that the weights are initially the same? A diagram which is not the torus At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together":   Is it so? I tried to glue them together, but end up with a normal torus. What does "no long range" mean on the soulknife rogue subclass mean? Recovering data from a faulty disk When I plug in a removable 1TB USB disk in my laptop, the operations led keeps blinking for a while and then you can hear the servo mechanism resetting with a short "tack".  I'd like to recover the data.    I believe there might be some way to get the data through low level IOs.    What I believe is that the mounting process assumes the disk is OK and I should prevent this before using ddrescue or something similar.  Does anybody have a solution to this problem? I am fairly sure that I understand the how Monte Carlo integration works but I am not understanding the formulation of how it is used to estimate Pi.  I am going by the procedure outlined in the 5th slide of this presentation   I understand the preliminary steps.  Pi is equal to 4 times the area of a quarter of the unit circle.  And the area of the top-right quarter of the unit circle centered at (0,0) is equivalent to the integral of the curve that is the top-right quarter of the unit circle in $0&lt;x&lt;1$ and $0&lt;y&lt;1$.    What I don't understand is how this integral is  $\iint I((x^2+y^2)&lt;1)P(x,y)dxdy$  where $P(x,y)$ is uniformly distributed in the unit square around the quarter circle (i.e. it is always equal to 1 if $0&lt;x&lt;1$ and $0&lt;y&lt;1$ and 0 otherwise).  So this would mean that  $I((x^2+y^2)&lt;1)P(x,y)$ is the function that is the top-right quadrant of the unit circle at $0&lt;x&lt;1$ and $0&lt;y&lt;1$ but I do not understand how this is true since the indicator function can only be 1 or 0.  I understand that it is probably written in this way to make Monte Carlo sampling easy (i.e. it is an expectation so just sample from $P(x,y)$ and get the average of the samples applied to $I((x^2+y^2)&lt;1)$) but it just does not make intuitive sense to me why that integral represents the area under that curve.  Could someone provide an intuitive explanation of this.  Maybe show how that integral was derived in a step-by-step way?  EDIT:  I was able to gain a better understanding by relating the expectation to an area.  I will explain it here in case it helps anyone.  First start with relating Pi to the area of the top-right quadrant of the unit circle  $\pi=4\times A_{tr}$  Then we place the top-right quadrant into the unit square.  And under a uniform distribution over the unit square, the area of the circle quadrant is proportional to the probability of obtaining a sample from it.  It follows that the following equality holds  $P(x^2+y^2&lt;1)=\frac{A_{tr}}{A_{square}}$  and $A_{square}=1$ so  $P(x^2+y^2&lt;1)=A_{tr}$  And substituting into the original equation  $\pi=4\times P(x^2+y^2&lt;1)$  and it is also true that $P(x^2+y^2&lt;1)=E[I(x^2+y^2&lt;1)]$ which is equal to the original double integral.  So I understood it by relating the area to a probability then relating that probability to an expectation that is equivalent to the integral.  Let me know if I have made any mistakes. Drift velocity of electron Windows 10 changes UEFI boot order every time I have an HP 500-413nl (a desktop) and Windows 10 keeps changing the boot order in UEFI/BIOS every time, so that if I want to boot Ubuntu 16.10 (64-bit) I always have to press F10, thus entering the BIOS setup and changing boot order…  Is there a workaround for my PC? What should I do? I'm a little desperate.  P.S.: I have already disabled "Secure boot" and "Fast/quick boot" in the BIOS and "Fast startup" in Windows options / Control panel. When should you push variables in quotes in boolean conditions? Are the following two boolean expressions the same?  if [ -n $1 ] ; then  if [ -n "$1" ] ; then   If not - When should you put a variable in quotes? How can I change media volume when no media is being played? Sometimes I want to play games in (near) silence, and don't like the background music to disturb people. If I use the volume buttons, the ringtone volume is changed instead of media volume. When I'm already in the game, the volume buttons do change the volume of background music, but it's kinda too late.  Is there a way for me to change the media volume to a level I like, from the home screen, before starting apps/games? In the PHP manual, to show the syntax for functions with optional parameters, they use brackets around each set of dependent optional parameter. For example, for the  function, the manual reads:  string date ( string $format [, int $timestamp = time() ] )   Where $timestamp is an optional parameter, and when left blank it defaults to the  function's return value.  How do you go about creating optional parameters like this when defining a custom function in PHP? How to restore Windows 7 MBR without a CD public static map&lt;id,RecordType&gt; MapRecordtype = new Map&lt;id,RecordType&gt;([SELECT id, DeveloperName FROM RecordType where sobjecttype ='move__c' and                                         Developername in('Assignment','Draft','Domestic_Draft','LTST_Draft','MX_Domestic','MX_Domestic_draft','Termination','International_Relo','Localization','J1',                                        'Back_To_Back','Repatriation','PAM','US_Domestic_Relo','CHN_Domestic_Relo','CHN_Domestic_Draft')])   Can we write schema for above query? For linear regression, we can check the diagnostic plots (residuals plots, Normal QQ plots, etc) to check if the assumptions of linear regression are violated.  For logistic regression, I am having trouble finding resources that explain how to diagnose the logistic regression model fit. Digging up some course notes for GLM, it simply states that checking the residuals is not helpful for performing diagnosis for a logistic regression fit.  Looking around the internet, there also seems to be various "diagnosis" procedures, such as checking the model deviance and performing chi-squared tests, but other sources state that this is inappropriate, and that you should perform a Hosmer-Lemeshow goodness of fit test. Then I find other sources that state that this test may be highly dependent on the actual groupings and cut-off values (may not be reliable).  So how should one diagnose the logistic regression fit? The school where I work has an S when it is an acronym (NEGS) it looks odd with an apostrophe to show ownership e.g. NEGS' tennis team...any thoughts? chown -R root.root / So I was working and there was a directory that I was periodically changing to my user in  a root terminal so that I could open the files in text editors, but I accidentally pressed / instead of . and this was the command I put out:  chown -R emil.emil /   Then immediately panicked ^C and put in another command, as I thought I could always change back the files I needed to access:  chown -R root.root /   So now I can't do su from emil to even try to fix this, as I'm getting permission denied error. I'm on Ubuntu could someone tell me if there is a way I could fix this, without absolutely reinstalling the whole thing.  TIP  If you are trying to get root access, and are getting this permission denied error when using su try this command:  gksu gnome-terminal   It will prompt you for the root password and open a new root terminal for some reason this works and gives me root access, when su doesn't.