I would like to know if someone can help me with this.  "Show that a finite ring $R$ with unity $1\neq 0$ and no divisors of 0 is a field."  The original exercise asked me to show that it was a division ring, it's done. It also says that showing that commutativity holds is hard. I wonder, is it too hard? If it isn't, how could I show that?
I'm relatively new to ring theory so this is probably a simple question.  Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.  I am wondering if we can rearrange these properties and always get implication or produce finite rings with only 2 of the above properties.  Explicitly my questions are:  1) Does a finite ring that is commutative with multiplicative inverses always have no-zero divisors? (EDIT: this one is pretty easy too, let $xy=0$ and assume $x\neq 0$. then $x^{-1}xy=x^{-1}0$. Hence, $y=0$ as desired.)  2) Is a finite ring that has multiplicative inverses and no zero divisors always commutative?  If these are actually simple exercises, I'd just like a hint to get started with the proof or any counterexamples.  Thanks :)
I want to typeset a piece of code containing line breaks.  New lines should indent to a specified point in the previous line.  Here is a monospaced example to illustrate what I mean.  Notice how 'case' and 'of' line up below, as well as 'let' and 'in':  swap : forall a, b. Tuple a b -&gt; Tuple b a swap a b x = case x              of tuple a b y z. let x' = tuple b a z y                                in x'   The code I want to typeset is not monospaced, so I can't just use a verbatim environment to get what I want.  The actual LaTeX of the first line, for instance, is  \mathsf{swap} : \forall\alpha,\beta.\;\mathsf{Tuple}\;\alpha\;\beta\to\mathsf{Tuple}\;\beta\;\alpha   How can I align some text to a specific point in the previous line? What are all the common undefined behaviours that a C++ programmer should know about?  Say, like:  a[i] = i++; After much trial and tribulation, I thought I had drush all ready to roar. Much excitement. But then, no joy...  Drush is working. The sites operate fine via MAMP. Regardless, a "drush cc all" command returns:   No Drupal site found, only 'drush' cache was cleared.                                          [warning] 'all' cache was cleared.                                                                       [success]   This question has stumped drupalcon attendees and very experienced developers. Any ideas?  I am on Mac OS 10.9.3 using MAMP (free) Version 3.0.5  Here is my drush status:   Drupal version         :  7.29-dev                                 Site URI               :  http://default                           Database driver        :  mysql                                    Database hostname      :  localhost                                Database username      :  db_user                             Database name          :  db_name                        PHP executable         :  /usr/bin/php                             PHP configuration      :                                           PHP OS                 :  Darwin                                   Drush version          :  7.0-dev                                  Drush temp directory   :  /tmp                                     Drush configuration    :                                           Drush alias files      :                                           Drupal root            :  /Applications/MAMP/htdocs/mysite.dev   Site path              :  sites/default       Have tried executing drush from /Applications/MAMP/htdocs/mysite.dev; from /Applications/MAMP/htdocs/mysite.dev/sites/default; and using drush cc all --uri=http://localhost/mysite.dev  Tried running drush from sites/default. 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I just found out that 3 light bulbs sockets in my home  are connected this way, it first came to my mind to fix it but then I recalled the flow of current from positive to negative is just a convention since the electrons are charged negatively and they actually flow from N to L so i cant really find an solid excuse to mess with the wiring for the next 2 hrs Need hint: Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent. Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is. How to hide users from the GDM login screen? I have recently added several new users, that I need for qmail. Now they appear in the box in the login screen and clutter it, and I have to scroll to find my user. How can I hide those users from the login box? 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