Let $R$ be a (not necessarily unitary), we know that if R doesn't have zero divisors then it satisfies the cancellation laws: $\forall a,x,y \in R$ with $a \neq 0$ $ax=ay$ implies $x=y$ and similarly,  $xa=ya$ implies $x=y$ but $xa=ay$ does not imply $x=y$ It is clear that this implication is true if $R$ is conmutative, so then I thought if the converse is true, but I could't either prove it nor disprove it. So the exact statement is: is a ring $R$ without zero divisors and for which $ax=ya$ hapens only when $x=y$ or $a=0$ necessarily commutative? Can you prove this or give a couterexample ring?
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$.   This is my proof to show that a ring is commutative if it has the property that ab = ca implies b = c.  We need to show that if x, y ∈ R then xy = yx. Let a = x, b = yx and c = xy. Then ab = x(yx) = (xy)x = ca. Thus, by the hypothesis, b = c, or xy = yx. Thus, for x, y ∈ R we have xy = yx.   How would I show when $a\neq 0$?
Why can "trans" be replaced with an x? I can't think of an example, so I may be wrong about this, but I think I've seen people replace the prefix "trans" as in transport with an x. "Cross" makes sense, as in "railroad crossing", and I guess "chris" is sort of a punny extension, the extension being like "chris cross" and the example being "xmas", but why trans? Why do electrons occupy the space around nuclei, and not collide with them? We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However...  How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons do collide, and, if so, what occurs? There is no such thing as magnetism? Here's an interesting "proof" that there is no such thing as magnetism. I know the answer but I love this so much I had to ask it here. It's a great way to confuse people!  As we all know, $$\nabla \cdot\vec{B} =0$$ Using the divergence theorem, we find $$ \iint_S \vec{B} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{B} \, dV = 0$$ Since $\vec{B}$ has zero divergence, there exist a vector function $\vec{A}$ such that $$\vec{B} = \nabla \times \vec{A}$$ Combining the last two equations, we get $$\iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Applying Stokes' theorem, we find $$\oint_C \vec{A} \cdot \hat{t} \, ds = \iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Therefore, $\vec{A}$ is path independent and we can write $\vec{A} = \nabla \psi$ for some scalar function $\psi$. Since the curl of the gradient of a function is zero, we arrive at: $$\vec{B} = \nabla \times \nabla \psi = 0,$$ which means that all magnetic fields are zero, but that can't be!  Can you see where we went wrong? What Is the Difference Between Wizard and Warlock in Potterverse? One of Dumbledore's titles is "Chief Warlock of the Wizengamot".   What is the difference between a Wizard and a Warlock in Potterverse? As we don't see the term Warlock used often in the books, my impression is that a Warlock is somehow officially different from a Wizard.  Please no Wikipedia, HP Wikia, or dictionary.com answers. I'm looking for a canon-based&#185; explanation.  &#185;Any of the ten HP books or a quote from JKR I have a dell laptop with 4GB of ram and 500GB of storage.I just finished installing windows 8.1 32bit version, which utilizes 3GB of ram.  Can I install and run Ubuntu to utilize that unusable portion of Ram.Then how?  Thanks in advance Underfull \hbox in bibliography I am getting lots of "Underfull \hbox" warnings that point to the *.bbl file. It looks like these messages appear only when URLs are very long, like in the below example:  @techreport{nistguidesec,   author    = {Wayne Jansen and Timothy Grance},   title     = {Guidelines on Security and Privacy in Public Cloud Computing},   month    = {January},   note     = {Draft Special Publication 800-144. Available at \url{http://csrc.nist.gov/publications/drafts/800-144/Draft-SP-800-144_cloud-computing.pdf}}, }   In the produced PDF text is correctly hyphened and links are correctly split over several lines (I am using hyperref package)  How can I avoid annoying warning messages about "underfull \hbox" pointing to bibliography? Convert UTC date time to local date time How can the pixel values of a raster image be extracted under a polygon fishnet? Dealing with splits in my ground plane Whenever I'm making a cheap, 2-sided PCB, I run into this problem:    Where I have signal lines (on the top layer here) running over top of a big fat power trace (on the bottom layer).  Now, this is obviously terrible because there is no good return path on the bottom side.  For issues like this, Hott consultants has a good recommendation:    However, I wonder if a solution like the one below is also acceptable.    My intuition tells me yes, this solution is ok.  Over the break in the ground plane, the current will hop over the vias and then back down to the lower plane.  Some back of the envelope math tells me that this adds a trivial amount to the loop area, and therefore a trivial amount to the return impedance.  Can someone tell me...   How right / wrong my intuition is here about using vias? If my intuition is wrong, why? Also, if my intuition is wrong, what is the correct solution in this sort of situation (2 layer, with power traces under signal)?   For the record, these particular signal lines aren't terribly fast (500 kHz), but I would like to know how to deal with this situation when I have signals in the 5 - 10 MHz range.  Edit  Below is a screenshot of about 60% of the board.  Maybe it will help with the critique Capacitor Voltage Rating How to find the voltage rating of a capacitor after knowing the capacitance value.  Example, For a power supply, If the input voltage is 60V and output is 5V. The input capacitor(C1) value is 100uf and output capacitor value is 1000uf.  What is the voltage rating for both the capacitors and how to find them. I have an inkling that the following word usage is grammatically-correct; however, it is immensely difficult to search for confirmation, as all I receive for queries like &quot;Proper as an adjective&quot; are reams of results on &quot;Proper adjectives&quot;. The word itself I am using as a postpositive adjective, similar in form to the phrase &quot;time immemorial&quot;.  The shelter remains in name only; the shelter proper is in another town. Work in theory remained, although operations proper had mostly been outsourced.  Can anyone advise? Sort order in Windows Explorer The behaviour described below occurs on Windows-7 systems and on Windows XP. We operate in a dual-language environment - English and Hebrew.  When in Windows Explorer we sort files by name, the order in which they are listed is not what we would expect. Here is a list of file names as sorted by Windows Explorer (all of the files have a .pdf suffix):   1G110033H-PP 19C050G-PP-ORB 19C050H-PPRM 19C100H-PPRM 19C-MBPS-PP 19C-MBPS-PP-1 29AAC050-PP 29AAC100-PP 29AAC100-PPUL   29B004064-PP 101AC050-PP 101AC100-PP 101B100-PPE 1091003G-PPFSUL 10108033G-PPSA 10125033H-PPM     It looks to me that first the items are sorted according to the position of the first alphabetic character in the name, and then, within those groups, they are sorted in "normal" alpha-numeric order.  That is, all the files with an alpha character in the first position are on top of the list, followed by those with the first alpha character in the second position, followed by those with the first alpha character in the third position, and so on.  An alternate way of looking at this is that, in a file name composed of numbers and letters, the sort treats the first group of numbers in the name as the major sort node, with the rest of the name being the secondary sort node.  Now that I understand the sequencing logic, it's not a big problem, but I was wondering why this happens? How do I assign the output of a command into an array? Thesis chapter headings, any suggestions? I've found a thesis with really beautiful chapter headings that I would like to use. What package am I going to need and how would I go about creating a heading that looks like the one below. Also, if you have or know of any nice headings do please share! I found this , but I am not sure if this is the same issue.  On , I did some reformatting of the indentation in the code block to make it a bit more readable and it was  for peer review. A few minutes later, it appears the OP made an edit that added some additional code. It looks like my suggested edit was applied when the OP saved his/her edit, and it looks like he got the "credit" for the edit.  I'm not really concerned about the 2 rep points; I'm more curious why this happened and if this is a bug (or limitation) in the editing feature, or if this is known/intended behavior?  Note that this is not a suggested edit that's overridden by a concurrent edit: the recorded edit starts from the post-suggested edit state (and in fact can be identical to it, as in  ). How do I hide a process in Task Manager in C#? I have a requirement to hide a process in Task Manager. It is for Intranet scenario. So, everything is legitimate. :)   Please feel free to share any code you have (preferably in C#) or any other techniques or any issues in going with this route.  Update1: Most of the users have admin privileges in order to run some legacy apps. So, one of the suggestion was to hide it in task manager. If there are other approaches to prevent users from killing the process, that would be great.  Update2: Removing the reference to rootkit. Somehow made this post look negative. How to rename multiple files by removing the extension? How will the super massive black hole affect our galaxy? US doctor working in Tripoli wants me to open online account How to resume baking fluid simulation? It happens sometimes that after finishing bake (e.g., for fluid simulation) we wish we could continue the simulation for two more seconds :(    How to use the cached bake to continue the simulation?  We played with the settings, but with no success.