It is possible to have a ring $A$,  with identity,  such that $A^m\cong A^n$ as left $A$-modules and with $m\neq n$?   I have that for commutative rings this is not possible, but I have no idea of the proof.
How can i show that two $R$-modules of finite rank are isomorphic if and only if they have the same rank, i.e., $R^n \cong R^m$ iff $n=m$.
Inverse of function is continuous or not? If $f:I\subseteq \Bbb R \rightarrow \Bbb R$ is injective and everywhere continuous in its domain then $f^{-1}$ is also continuous everywhere. Here  $I$ cannot be discrete set and $f$ is single variable function The above statement is always correct or not? I thought it is always correct. Am I correct or not? Fractions with large elements What is best practice for typesetting fractions with large objects in the numerator and denominator?  For example, fractions with limits, integrals, summations, etc... cause the expressions to look contracted and somewhat odd.  I understand that it is probably best to leave this as is when writing mathematics inline, but as a stand-alone fraction I feel it would be much less distracting if the symbols were typeset at normal size and the fraction was given more height in which to place itself.  Are there standards (for pure mathematics) regarding this problem? And what are the options - if it's generally permitted - for expanding the fraction size to incorporate large objects?    Example  \frac{\sum_{i=0}^{n} i^2}{\iint_{D}\frac{\partial (x,y)}{\partial (u,v)}\,\mathrm{d}u\,\mathrm{d}v}   Both the summation and the integral look somewhat cramped. Using ReplaceAll on SparseArray This is similar to , except it's a bug report, not a feature request.  Currently, the nominee's comments on his/her own post are not highlights, unless you click "add / show more comments"  When the page is loaded it looks like:    AfterI click "add / show more comments", it becomes:    Could we have the blue highlighting on the post owner's comments there all the time? It becomes difficult to see the nominees responses to questions, especially on nominations where there aren't enough comments to enable the "add / show more comments" features, like Matt's. Where does the question mark go — inside or outside the parentheses? I recently read the .  This book had the idea of the time you live as something that can be used to pay for things. When you run out of "money" you die.  I was wondering is this the original source for this idea? Or does anyone know of an earlier book with the same idea? Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}s\,H(s)\right)\tag{1}. $$ In Breuer and Petruccione, it's said that if the commutator of the Hamiltonian at some time $t$ with itself at some other time $t^\prime$ is a $c$-number function/ a complex function, i.e. $$\left[H(t),\,H(t^\prime)\right]=f(t,t'),$$ then the time evolution is given by $$ U(t)=\exp\left(-\frac{1}{2}\int_0^t\mathrm{d}s\int_0^t\mathrm{d}s^\prime\,\left[H(s),\,H(s^\prime)\right]\Theta(s-s^\prime)\right)$$ $$\times \exp\left(-\mathrm{i}\int_0^t\mathrm{ds}\,H(s)\right),\tag{2} $$ where $\Theta(s-s^\prime)$ is the Heaviside function. No proof or reference is given and I could not find any explanation anywhere, this is why I ask here. Any help would be very appreciated. Windows 10 update broke my MBR / GRUB How to see time stamps in bash history? Is there any way I can see at what time the commands were executed from the bash history? We can see the order but is there any way I can get the time also?  Bottom-Line: Execution time in the Bash history Why are we in love "with" someone? How to convert from one grading scheme to another? I'm trying to figure out the equivalence between grade schemes in different countries.  According to some Google results:   If you are a U.S. citizen, you need a GPA of 3.2+ (it varies, some say 3.2 and some others say 3.6, etc) to apply for a position in the UK (job or university) where the prerequisite is to have a minimum grade of 2:1. In UK 2:1 is earned if you have a 60-69% of the points. I've read that 2:1 is the same as 67%. In Spain we use a grading scheme of 10 points. So according to (2), for have a UK 2:1 you need to have 6-6.9 points. If you are Spanish and have 6.7, when converting it to GPA the result is 1.5.   However, if you take the above points and treat them as a math equation, the final result will be GPA 1.5 = GPA 3.2, which doesn't make any sense.  GPA 3.2 = UK 2:1 UK 2:1 = Spain 6.7 Spain 6.7 = GPA 1.5   Replacing now...  UK 2:1 = GPA 1.5     GPA 3.2 = GPA 1.5   My question is if someone has a better explanation on converting grades obtained in different countries. Google Image Search using Android I have an image on my smartphone and I want to paste it into Google Image Search to see if I can get similar images.  I can do this by pasting the image using desktop but not using Android ...  Is there such a possibility? Is circular reasoning always a fallacy? If I want to know the meaning of wget -b, I see the manual by man wget, then search the -b option.     -b    --background        Go to background immediately after startup.  If no output file is specified via the -o, output is redirected to wget-log.   I want to get the result by a command like man wget -b. (Of course this doesn't work.)  Is there a similar way to make it possible? Emulate "double" using 2 "float"s $\require{enclose}\enclose{horizontalstrike}{\rm Greatest}\!$ Least prime factor of $n$ is less than square root of $n$, proof I remember reading this somewhere but I cannot locate the proof. I'm trying to understand Python's approach to variable scope. In this example, why is f() able to alter the value of x, as perceived within main(), but not the value of n?  def f(n, x):     n = 2     x.append(4)     print('In f():', n, x)  def main():     n = 1     x = [0,1,2,3]     print('Before:', n, x)     f(n, x)     print('After: ', n, x)  main()   Output:  Before: 1 [0, 1, 2, 3] In f(): 2 [0, 1, 2, 3, 4] After:  1 [0, 1, 2, 3, 4] Why is my iron ore disappearing when mined with a wooden pickaxe? I am on Peaceful mode, and I am using a Wooden Pickaxe. Whenever I mine Iron Ore, it just disappears. Am I doing something wrong? What is the unit of the coordinate system in PSTricks? If I use the following code, there two different kinds of units. One for the coordinate system used to declare \begin{pspicture} with (0,0) to (20,10), and the centimeter used in 4.2cm.  \begin{pspicture}(0,0)(20,10) \rput[lt](0.5,9){\psframebox[linewidth=1pt,framearc=0]{\parbox[c]{4.2cm}{\raggedright Text Goes Here...}}} \end{pspicture}   Can I use a unified dimension for both PSTricks objects and \parbox?