Let $R$ be a nonzero commutative ring, and let $n,m$ be integers. Let $f:R^n\rightarrow R^m$ be an injective homomorphism of $R$-modules.  I'm trying to show that $n\leq m$.  My idea: Assume that $n&gt;m$, let $i:R^m\rightarrow R^n$ be the "embedding in the first $n$ coordinates", and take the composition $g:R^n\rightarrow R^n$ define by $g=i\circ f$.  First, $g$ is injective. Second, we can apply a version of Nakayama's lemma to  get a monic polynomial $p(X)\in R[X]$ such that $p(g)=0$.  Now, I somehow want to use the fact that the image of $g$ consists of elements  which are zero on the last coordinate (at least), but I'm circling around this with no success.  This is an exercise from Atiyah-Macdonald.
I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold.    This is exercise 2.11 from Atiyah and MacDonald's Introduction to Commutative Algebra, but unfortunately all the solutions available online are either very sparse with regard to this question or seem to use (to my surprise) a generalized version of Cramer's rule. Is there perhaps some other "cleaner" approach to solving this problem?
Everyone I speak to says that a 50mm lens is the best and beats them all as an all-rounder. I was wondering whether to get myself a decent, fast 50mm as well. However I'm still not 100% sure whether I really need such a lens.  I use a Pentax K10D and mainly shoot people (kids, portraits, some street shots). The lens I love and use most is the DA 70mm f/2.4 Limited. I often also use a DA 40mm f/2.8 Limited. They are great little lenses that produce very high quality images but they could be a bit faster.  Is is worth spending 400 quid on a Sigma 50mm f/1.4 in my situation? It seems like a good compromise between the 40mm and 70mm and much faster (in fact, 2 f-stops faster than the DA 70mm f/2.8!). On the other hand would the lens be sufficiently different from my current lenses? I have seen that the new  shows off H.265/HEVC playback, the successor of H.264. I wonder how I may get this support in Linux. Is there a need to install a codec specific for this format? How? How do I find out which process is locking a file using .NET? Including +/- signs to battery in Circuitikz I'm attempting to draw a battery (using the battery1 component) that includes positive and negative signs at the appropriate ends. A  that was answered here (using version 2.4) shows exactly what I mean:    Only, of course, I want to be using a battery1 symbol instead. So, I have my code as follows:  \usepackage[americanvoltages,fulldiodes,siunitx]{circuitikz} \begin{document} \begin{circuitikz}     \draw (0,0) to [battery1=\SI{9}{V}] (3,0)                 to [pC,v=$v_C$] (6,0); \end{circuitikz} \end{document}   And I end up with this:    Notice the lack of +/- signs (I know how circuits work so I don't need to be actually shown these signs, but for the sake of copying my class notes diagrams I want these signs to be included in my latex diagram).  In either case, I think it may have something to do with the version of Circuitikz I'm using because I'm using the latest version (I believe, 0.3.0) but in the previous question linked above, they were using 2.4 and it worked out for them.  I would really like to find a way to add +/- without having to manually add it with Tikz or something, i.e. just have it naturally part of the component without having to add nodes or extra Tikz code etc. If that's not possible, then I'll have to settle for that as a last result.   Thank you for your help!  EDIT: I should also mention that I've tried using battery1, v=$9V$ instead of battery1=\SI{9}{V} and to no avail (the +/- signs still don't show up). keyboard hard remap keys? Intuitive Understanding of the constant "$e$" We're injecting records into a journey using a Salesforce data event.  I also want to advance the people in the journey based on updates that happen in Salesforce (e.g., as an Opportunity record advances through the sales process), but I can't figure out how to do that.  I've read all the documentation I can find, but there doesn't seem to be a Journey Builder widget that will wait to be prompted by an event in Salesforce before continuing someone on a journey.  At the moment, I'm completely stuck.    I'd greatly appreciate any advice or information on how I can make journeys event-driven in this way.  Thanks very much! Migration stats not consistent between sites Apparently, as a result of  it appears that Yannis and myself see different statistics when going to the programmers.SE.com migration stats tool and the stackoverflow.com tool respectively.    :        while I see:        Why are the rejection rates so radically different?  Which is correct?  (Or am I simply misinterpreting the data somehow and these are representing different statistics) I have the following code for a Figure with 4 subfigures:  \documentclass{article} \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption}  \begin{document}  \begin{figure}[b]     \centering     \begin{subfigure}[b]{\textwidth}         \includegraphics[width=\textwidth]{./plots/arm1.pdf}         \subcaption{$Q^{*}$ values for arm 1}         \label{fig:arm1}     \end{subfigure}     ~     \begin{subfigure}[b]{\textwidth}         \includegraphics[width=\textwidth]{./plots/arm2.pdf}         \subcaption{$Q^{*}$ values for arm 2}         \label{fig:arm2}     \end{subfigure}      ~     \begin{subfigure}[b]{\textwidth}         \includegraphics[width=\textwidth]{./plots/arm3.pdf}         \subcaption{$Q^{*}$ values for arm 3}         \label{fig:arm3}     \end{subfigure}      ~     \begin{subfigure}[b]{\textwidth}         \includegraphics[width=\textwidth]{./plots/arm4.pdf}         \subcaption{$Q^{*}$ values for arm 4}         \label{fig:arm4}     \end{subfigure}      \caption{$Q^{*}$ values for different arms}     \label{fig:arms} \end{figure}  \end{document}   This results in   But this way, the 4 subfigures are too large to fit on 1 page. I would like to split this over 2 pages, keeping the subcaptions a, b, ... How can I do this? Assign multiple keyboard shortcuts for one action Functions of Independent Random Variables OK guys I have this problem:  For $x,y,p,q&gt;0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$  It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the solution? raw data stationary but still can see trend and seaonality is stl I want to describe my business and want to refer to it in a sentence.  I am trying to say:      "In 2015 we established a business (that is) named X".   Intuitively, the "that is" part is redundant, and makes the sentence more cumbersome. But I cannot think of a reason why we should be able to drop the verb.  Is the version without the "that is" part grammatically correct? How can it be explained? A bunch of CSS files were pulled in and now I'm trying to clean things up a bit. How can I efficiently identify unused CSS definitions in a whole project? After reading this question about  and asking myself two questions about  and , I realize that these activities are for showing what you found, make the audience be attracted. If they need further explanation, they should read your paper, which has successfully defensed and proved that your work is validated. One important thing when you attempt to attract people is concentrate on the important, which means unnecessary things need to be trimmed. I think the reference is one of those. So, should we put the references in slides/poster when we already have them in paper? Linux command line. Move all files and directories in directory, except some files and directories I have a folder A which has files and directories, I want to move all those files and directories to another folder B, except file, file2, directory, and directory2.  How can this be done? What can I do if a repository/PPA does not have a Release file? When updating, I get the following error message:   W: The repository 'http://ppa.launchpad.net/mc3man/trusty-media/ubuntu xenial Release' does not have a Release file.   Here, I find another statement on this error:   This recommends removing certain PPAs; and, I'm not sure if I should do that since it might mean not getting the updates that I need.    Is this what I should do? align vs equation I always use align in my documents, and avoid equation. Is there anything wrong with that? My reasoning behind this: align > equation, so why not use it? I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.  Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by :     New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells.