I know that there is definitely module and vector space these two objects are similar. see the quotation below:      A module is abstractly very similar to a vector space, although in modules, coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. A module taking its coefficients in a ring is called a module over , or a R-module.   Question 1: I did not understand the meaning of line "general algebraic objects " in the above paragraph.  Question 2: What are other things that are similar or not similar  between module and vectors space in terms of geometry?
Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind.
TZInfo::DataSourceNotFound error starting Rails v4.1.0 server on Windows I can't really remember how long ago it was, but I think the book was marketed towards children/teens.  All I remember is the basic premise of the story.  Some story about these unusual little blue electric critters that only certain people could see, and these kids took it upon themselves to go about destroying them since they fed off energy or something like that.  Not sure whether they fed off people or off electricity... or both maybe?  I think the dominant method of destroying them was an electric baton that was taken from law enforcement.  I wanna call them buzz batons, but that's probably thanks to Artemis Fowl. Finding the limit of $(1-\cos x)/x^2$ How to calculate Area Under the Curve (AUC), or the c-statistic, by hand I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. For example, here are top five accounts in each site (this picture was taken when I was in Biology SE).    As you can see, all the badges are DNA, while Academia's badges should be hats, so on.  What do you think? So I'm using an app that stores images heavily in the DB. What's your outlook on this? I'm more of a type to store the location in the filesystem, than store it directly in the DB.  What do you think are the pros/cons? ''More than one row returned by a subquery used as an expression'' postgres error How to Configure Launcher Placement in Multi-monitor Setup I use a large monitor as primary and laptop screen as secondary. Is it possible to move the launcher to my big monitor or do I have to turn off my laptop monitor?   "Monitor Preferences" (Launcher search -> "Monitors") does not have a way to set the default/primary monitor. nvidia-settings which I just installed says "You do not appear to be using the NVDIA X driver. Please edit your X configuration file (just run nvidia-xconfig as root), and restart the X server.  sudo nvidia-xconfig says sudo: nvidia-xconfig: command not found.  I guess I don't have an AMD driver because I don't have "AMD Catylist Control Center" in the launcher search. What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? \documentclass{article} %\usepackage{amsmath} It does Not work ! \begin{document} \[ x\lt y \gt z \] \end{document}  What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? Edit I am using MathJax library in my site under Asp.net Mvc 3. Web application is vulnerable to XSS and XSRF attacks so &lt; and &gt; should be avoided as the input from the users. My database will save LaTeX commands that can be rendered by both LaTeX compiler and MathJax. Therefore, I need \gt and \lt that already work in MathJax. Source:  Rendered: I'm trying to solve an exercise about first order stochastic dominance, and there is this one question that I'm not able to answer.  Here are the hypothesis of the exercise:  Let Y ∼ F and Y' ∼ G, that means Y (resp. Y') has a probability law given by the distribution function F (resp. G).  Assume that Y ≥ 0 and Y' ≥ 0 and E(Y) + E(Y')&lt; ∞.  Here comes the question:  -Show that if the survival function of Y first order dominates over the survival function of Y', then if Y and Y' are independent, we have P(Y ≥ Y') ≥ 1/2.  I don't see how to use that Y and Y' are in this case independant.  Maybe I should set X=Y-Y' and then use the convolution?  Any help, would be appreciated, thanks !  Alex PHP function to build query string from array I'm looking for the name of the PHP function to build a query string from an array of key value pairs. Please note, I am looking for the built in PHP function to do this, not a homebrew one (that's all a google search seems to return). There is one, I just can't remember its name or find it on php.net. IIRC its name isn't that intuitive. What is the rule to convert the 'mm' notation to the 'optical zoom' notation? I searched a bit and found this one:  optical zoom = maximum focal length / minimum focal length   For example a 18-55mm lens would have a 3x optical zoom, and a 18-200mm lens would have a 11x optical zoom. Is that right? BJT Astable Multivibrator How to get the Input and output in a terminal to a text file? I want to create a text file which will add each command I am going to run on the terminal as a new line entry in the text file. i.e., I need to keep a record of what are the things I'm doing in the terminal. Further if I can keep a record of outputs I'm getting in the terminal, along with the inputs in the terminal, as a text file, it will be much better.   Each time I forget a command I have to search again in the Internet for the command.  Can I can make a file like above? You require a reputation of 50 to leave comments on others' questions/answers; the FAQ simply states this as a fact but does not explain it. I'm a new user and I have identified this as an issue very quickly through my own experiences so far. Looking at other questions seems to also provide justification against this 50 reputation requirement for comments on others' questions/answers.      In my own experience, I've seen a few questions like "how do I do x?" then the same poster has answered with "in the end I did it like this: ..." where their solution was perfectly sensible, but there may be some gotchas to watch for - in this case I want to vote up their answer and/or comment on it to point out the potential gotchas, but I can do neither!  To me all of the above seems justification against this rep requirement to comment, so what is the counter-argument and justification for it? I want to insert something into a STL list in C++, but I only have a reverse iterator. What is the usual way to accomplish this?  This works: (of course it does)  std::list&lt;int&gt; l; std::list&lt;int&gt;::iterator forward = l.begin(); l.insert(forward, 5);   This doesn't work: (what should I do instead?)  std::list&lt;int&gt; l; std::list&lt;int&gt;::reverse_iterator reverse = l.rbegin(); l.insert(reverse, 10); What is the best way to explain how to choose between "its" and "it's"? How long should it take to refresh a Developer Sandbox? I refreshed a Developer Sandbox, and almost 24 hours it is still "pending". Is this normal? Are there any actions I can take to speed up the process? When I created the Sandbox it took barely more than 10 minutes to create.  Thanks, Adrian Make a div fill the height of the remaining screen space Ball $X^*$ is weak-star metrizable iff X is separable in Conway's book. This is a Theorem in Conway's book     I don't understand in the third paragraph why there are open sets $\{U_n:n\geq1\}$ in $(ball X^*,wk^*)$ such that $0\in U_n$ and $\bigcap^{\infty}_{n=1}U_n=\{0\}$. By the definition of the relative weak-star topology on ball $X^*$, for each $n$, there exists a finite set $F_n$ contained in $X$ such that $$\{x^*\in\text{ball }X^*:|\langle x,x^*\rangle|&lt;1 \text{ for all }x\in F_n\}\subseteq U_n.$$  Can someone explain this for me？ Thank you so much!!