If M is a free module over a commutative ring, then any 2 bases for M will have the same cardinality. I'm looking for an example of a module that doesn't have this property, as well as examples of the bases in question.
The usual example of unitary ring without the  property is the ring of column finite matrices, and in this case we have $R\simeq R^2$ as (left) $R$-modules. (See also .) In particular, we have $R^2\simeq R^3$.      I wonder if there is an example of unitary ring (without the IBN property) such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules).
I need two bib-like sections: a "References" giving works explicitly cited in my text, and a "Bibliography" (or "Further reading" if you prefer) giving everything from my bibfile that didn't already appear in "References". The first is of course completely standard, but the second... I'm stumped.  Further loose constraints: I'm somewhat committed to biblatex and the biblatex-apa style. But I'll happily accept an answer not related to them if there's any chance I can adapt it. Why do I have to press Ctrl+D twice to close stdin? Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. In a USB cable, is it OK to swap the D+ and D- wires? I heard that D+ and D- are differential signals, does it matter if I swap them when connecting a USB device to the computer? Execute script after specific delay using JavaScript Is there any JavaScript method similar to the jQuery delay() or wait() (to delay the execution of a script for a specific amount of time)? How can one set the executable bit on files using umask? According to documentation, a umask of 000 should work, but in practice this only works on directories.  Also open to other methods of automatically setting +rwx on a particular user's created files if you have any. How critical are macOS Mojave compatibility requirements? As per Apple support document, :     1. Check compatibility      You can upgrade to macOS Mojave from OS X Mountain Lion or later on   any of the following Mac models. Your Mac also needs at least 2GB of   memory and 12.5GB of available storage space, or up to 18.5GB of   storage space when upgrading from OS X Yosemite or earlier.         MacBook introduced in early 2015 or later   MacBook Air introduced in mid 2012 or later   MacBook Pro introduced in mid 2012 or later   Mac mini introduced in late 2012 or later   iMac introduced in late 2012 or later   iMac Pro (all models)   Mac Pro introduced in late 2013, plus mid-2010 or mid-2012 models with a recommended Metal-capable graphics card.      I am currently running macOS High Sierra on my Early 2011 MacBook Pro.  What does Apple's requirements mean for my ability to install Mojave? Is it possible or worthwhile to do so? Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same? Where is log file from rc.local? I have some commands to in my rc.local. I know that they are failing. How can I get log file with messages produced by executing rc.local? Where is it located?  I have checked the /var/log/boot.log. I know my messages are not there because I know already what is the reason of failure. But I still want to make sure from log file.  Note, I don't want to run script again, I could but I don't want. I would rather analyse wht happened during startup.  Thanks for any help.  Ubuntu 12.04 Desktop (if it matters) How to add flavouring ingredients to steamed or boiled veggies? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? How to apply a bevel modifier, so that it affects only specific edges? How to make the tick thickness as the axis line? I am trying to make the thickness of the ticks equal to the axis line thickness. That is what I have so far:  \documentclass{article}   \usepackage{pgfplots}  \pgfplotsset{compat=1.5}   \begin{document}     \begin{tikzpicture}     \begin{axis}[         xlabel=Binding Energy (eV),         ylabel=Intensity (a.\,u.),         xtick pos=left,         x dir= reverse,         ytick = \empty,         every x tick/.style={color=black, thin},         tick align=outside,         xlabel near ticks,         ylabel near ticks,         xmin = 96,         xmax = 108     ]         %\addplot[mark=none] table[x=BE,y=cnts] {si2p.txt};         % This data looks similar:         \addplot[mark=none, smooth] table[x=BE,y=cnts] {             BE      cnts             95      1             98      1             99      10             99.5    5             100 6             101 2             108 2         };      \end{axis}     \end{tikzpicture} \end{document}     But if you take a closer look on the ticks on the outside, you'll see that they are somehow clipped or a least thinner. They are also black so the style seems to work for them.    Is this a bug or can I change the clipping region? And as a second question: is thin the correct value for the thickness? In the Final Battle, when you remove doom tokens for rolling successes against an Ancient One do you stop counting successes for that roll? In Arkham Horror you have to roll a cumulative number of successes equal to the number of players to knock off a doom token from the Elder God's health. If you reach the number of players worth of successes do you then stop counting for the player and move on or do you count the extra successes against the next doom token?  We played it so that you kept on adding them but it meant that we turned Nyalothotep into mince meat quite quickly (One Character was blessed had a fight of 6 a shotgun and Fight +1 skill card though) Forgot which email I used, can't login I forgot what email I used for my other ploxtic id: how do you find out what my email was?   The account is user:2078229. Adding newmode with Xrandr - "800x480_60.00" I have 5-inch a 800x480 tft connected to the LVDS connector on a Portwell WADE-8020 mini-ITX embedded system board. I would like to build a XMBC/Kodi box for streming my music from my freenas PC with the tft showing what is been played, I got as fare as this...   sudo xrandr --newmode "800x480_60.00"  29.50  800 824 896 992  480 483 493 500 -hsync +vsync`enter code here   then,  sudo xrandr --addmode LVDS1 800x480_60.00   now that I get the correct resolution, I now have a black band on the top of the screen.  sudo xrandr --output LVDS1 --set "scaling mode"  "Full aspect"   However this did not solve the issue, also I know I need to amend the /bin/sh file with this from googling so I don't need to keep doing this at every boot. But I don't know how to as I'm very new to Linux.  #!/bin/sh xrandr --newmode "800x480_60.00"  29.50  800 824 896 992  480 483 493 500 -hsync +vsync xrandr --addmode LVDS1 800x480_60.00 Calculating lat/lons from PostGIS table which has projected coordinate system? I have a PostGIS table in a projected coordinate system (EPSG: 28355 or MGA94 Zone 55). In QGIS (with the PostGIS manager plugin) I calculate a coordinate by performing the following SQL:  UPDATE gisprod.buildings SET easting = x(centroid(the_geom))    Now I want to add a "lat" and "lon" field to this table and calculate the values using the appropriate coordinate system, in this case that would be (EPSG:4283 or GDA94).   Is there a way I can specify in the SQL statement to calculate the x/y coord using a different coordinate system? Does the cross-entropy cost make sense in the context of regression? Does the cross-entropy cost make sense in the context of regression (as opposed to classification)? If so, could you give a toy example through TensorFlow? If not, why not?   I was reading about cross-entropy in  by Michael Nielsen and it seems like something that could naturally be used for regression as well as classification, but I don't understand how you'd apply it efficiently in TensorFlow since the loss functions take logits (which I don't really understand either) and they're listed under classification Start a longtable in Latex at top of next page I want to use the longtable environment for a table. However, I do not know how to place it at the top of the next page, as I can do with the table environment. Do you know how to do it? Can I hire my kid and start a Roth IRA for him? I own an LLC, of which I am currently the sole member. It's my main source of income.  I am wondering if there is a reason why I could not do the following:   Hire my child as a part-time employee of the LLC, for an annual salary of $5500. Start a Roth IRA for the kid and put the $5500 into it. Deduct the $5500 as a payroll expense for my LLC, thereby reducing my amount of taxable income.   To me, this seems attractive because I'd lower my LLC's tax liability. Meanwhile, my kid would not have to pay any taxes on the $5500 because it is lower than the standard deduction. And by putting the money in a Roth IRA, he'd get an investment that would grow tax-free. It would essentially be like getting the pre-tax benefits of a traditional IRA with the post-tax benefits of a Roth.  The way I see it, I'd win, my kid would win and the IRS would lose, for once.  Is there any reason this wouldn't work? There is no minimum age or income for contributing to a Roth IRA. So as long as I could legally hire my kid, I don't see any obstacles.