Show that every finite integral domain is a field.  I have shown that every nonzero element in a finite ring with an identity is either a zero divisor or a unit.
This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:   let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,  then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$,  therefore by canceling $x_j$ we get $x_1x_2\cdots x_{j-1}x_{j+1}\cdots x_k=1 $,  by commuting any of these elements to the front we find an inverse for first term, e.g. for $x_m$ we have $x_m(x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k)=1$, where $(x_m)^{-1}=x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k$.   As far as I can see this is correct, so we have found inverses for all $x_i\in R$ apart from $x_j$ if I am right so far. How would we find $(x_{j})^{-1}$?
I have start point and end point of a polyline. How can I get a point on that polyline from end points specified by the given distance.    How can I get this by using arcpy provided that this script should work on ArcView without importing any modules (). I'm trying not to hard code values.  Related Question: For US federal tax, whether a taxpayer is subject to AMT is determined by a complicated way in Instructions for Form 1040, Line 45, which involves an incomprehensible "worksheet to see if you should fill in form 6251".  By simplification and approximation, is there a rule of thumb for an estimate of a salary value of a taxpayer, so that a salary above the value might cause the taxpayer to pay AMT with a big possibility, and a salary below the value might not with a big possibility?  Why do the following two sources seem to differ greatly in estimates of the boundary salary value:    says that  the possible boundary salary value for a single taxpayer  to pay AMT is $53,900.  says that      Taxpayers with small incomes fall within the AMT exemption limits.   That leaves the upper middle class to face the tax, especially those   with incomes between $200,000 and $500,000    Thanks. Is there a specific order to complete Saints Row 2? How Do I Allow A Standard User to Install Programs? Yes, I realize that defeats the purpose of standard user. But here is the situation. A friend of mine wants to set up time limits for his home schooled son, and his son needs to be a standard user in order to use Family Safety. But at the same time, his son (who I personally think is way too old to have time limits), needs to be able to update and install some of his games on his computer. So I guess it's just sort of "light" parental control (just something to help him focus and not get distracted with hours of gaming).  It is the basic version of Windows 8.1, so there is no Group Policy editor (gpedit.msc). However, I'm fairly good with computers and I figured out how to install it. But it doesn't have a lot of the features that the "real" gpedit seems to have. Here is an example of what is in my gpedit: How to add current date to cell if the rest of the cells (or some cells) in that row are edited, but only the first time? I have a column A4:A (heading = Date), and other columns after it B4:O.  I want column A to fill in today's date automatically when I fill in any other columns (some of these B4:O maybe empty, but most will be filled) in that row. But I only want that to happen the first time I edit any other columns within that row.  What I've tried:  A7: =IF(B7 &gt; 0,TODAY(),) Sorry if this is the wrong place to ask this, I'm new to the site :/  My issue is this: when editing in vim, my syntax file highlights underscores and carets when found in non-math text, which is fine, except vim doesn't recognize that "align" is a math environment. What do I need to do to fix this? Why we usually do not exceed ARMA(5,5) models in practice? Is there any mathematical justification for this? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. How can I query the name of the current SQL Server database instance? It is a bit of a "chicken or egg" kind of query, but can someone dreamup a query that can return the name of the current database instance in which the query executes? Believe me when I say I understand the paradox: why do you need to know the name of the database instance if you're already connected to execute the query? Auditing in a multi-database environment.  I've looked at all the @@ globals in Books Online. "SELECT @@servername" comes close, but I want the name of the database instance rather than the server. How can I donate to Ubuntu without PayPal? When I wanted to download Ubuntu 12.10 from ubuntu.com, I was asked to make a donation. However, there is only a PayPal option to make the payment. But I don't use PayPal. Is there any other option to make the payment? A significant amount of zombies were detected in your city Can anyone please help and tell me if this sentence is correct?  &quot;A significant amount of zombies were detected in your city.&quot;  In my opinion it should be &quot;A significant amount of zombies was detected in your city.&quot; because &quot;amount&quot; should be in agreement with &quot;was&quot; even though amount usually refers to something plural. Are both (was and were) acceptable in this case? Edit1: Yes, &quot;number&quot; would be the correct one in this case. So allow me to ask one more thing.  &quot;A significant amount of purchases were detected...&quot;  Does the use of &quot;number&quot; apply in this case also? As purchase is not exactly a thing/person. And if no, is it correct to say &quot;A significant amount of purchases were detected...&quot;? Edit2: So, please tell me what is the correct phrasing?  &quot;A significant amount of purchases were detected...&quot; &quot;A significant amount of purchases was detected...&quot;  Or, since purchases are countable, is the use of &quot;number&quot; instead of &quot;amount&quot; correct here, as in:  &quot;A significant number of purchases were detected...&quot; &quot;A significant number of purchases was detected...&quot;  Or would amount be correct since a purchase involves money and it refers to the amount of money? Would greatly appreciate an answer, and if possible a detailed explanation. How do I copyright my website? My website has been in development for a long time and cost a lot of money.  How do I go about securing an enforceable copyright on my website design and what is the proper way to post a copyright notice on the site so others know the work is copyrighted? I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? I just finished backing up my system and when I try to upgrade to Utopic Unicorn, I get the error "Cannot Calculate the upgrades." I ran the upgrade by sudo apt-get dist-upgrade, sudo do-release-upgrade after the GUI method failed, they all gave the same error.  The feedback gives some recommendations:  This can be caused by:  * Upgrading to a pre-release version of Ubuntu  * Running the current pre-release version of Ubuntu  * Unofficial software packages not provided by Ubuntu    I am not upgrading to a pre-release version nor coming from one, I'm going from Ubuntu 14.04 LTS to 14.10. I have no broken packages and I have unchecked all my non-official software sources in the 'Software &amp; Updates' manager.  My guess left is the Unofficial software packages not provided by Ubuntu. If anyone could give any help as to how to check for unofficial packages using Synaptic or another, my appreciation.  Are there any other factors that may be causing this error and how would I go about checking/fixing those?  Thanks.  Edit: I have no broken or held packages. Everything is up-to-date, and restarting everything or anything has no effect. How can I create a product key for my C# application? How can I create a product key for my C# Application? I need to create a product (or license) key that I update annually.  Additionally I need to create one for trial versions.  Related: I am using field image to insert image into article. The image can't show in article body.   How do I insert the image into article? WP List Table custom quick edit box - post meta data missing and columns change on submit I have extended class WP_List_Table to create a custom sortable table of posts along with a quick edit box with custom meta boxes.  The meta boxes populate and save but when I click submit the field values disappear and the defaults ( cb, title, tags, cats, etc) which are not even included in my class get added back to the column saved. This throws off the other columns on the page.  The saved custom field values return when the page is refreshed.  It seams like on submit the $post_object is missing.  I'm using a hacked get_inline_data function that includes the meta_values in my quick edit form instead of calling get_inline_data in my class.  My save function for the meta boxes:  add_action( 'save_post', 'save_inline_edit_meta' ); function save_inline_edit_meta( $post_id  ) {     global $pagenow;      if ( $pagenow == 'admin.php' ) {      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'standinghead' ] ) )         update_post_meta( $post_id, '_wnd_standing_head', $_POST[ 'standinghead' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'headline' ] ) )         update_post_meta( $post_id, '_wnd_alt_title', $_POST[ 'headline' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'deck' ] ) )         update_post_meta( $post_id, '_wnd_deck', $_POST[ 'deck' ] );     }   return $post_id;  }   My Javascript save function:  save : function(id) {         var params, fields, page = $('.post_status_page').val() || '';          if ( typeof(id) == 'object' )             id = this.getId(id);         $('table.widefat .inline-edit-save .waiting').show();         params = {             action: 'inline-save',             post_ID: id,             post_type: 'post',             edit_date: 'true',             post_status: page         };         fields = $('#edit-'+id+' :input').serialize();         params = fields + '&amp;' + $.param(params);          $.post('admin-ajax.php', params,             function(r) {                 $('table.widefat .inline-edit-save .waiting').hide();                  if (r) {                     if ( -1 != r.indexOf('&lt;tr') ) {                         $(inlineEditPost.what+id).remove();                         $('#edit-'+id).before(r).remove();                         $(inlineEditPost.what+id).hide().fadeIn();                     } else {                         r = r.replace( /&lt;.[^&lt;&gt;]*?&gt;/g, '' );                         $('#edit-'+id+' .inline-edit-save .error').html(r).show();                     }                 } else {                     $('#edit-'+id+' .inline-edit-save .error').html(inlineEditL10n.error).show();                 }             }         , 'html');         return false;     }, What would happen if two wizards made each other a Horcrux? Would they be invincible? How would you kill one of them? I have an area which is divided into several zones the borders of which I have defined as separate layers.  The layers are defined by sql queries on a postgis table of line segments joined with a table that says which segments are part of which zone boundaries.  What I want to do is produce a set of maps where each map is a zoom to layer extent for each zone and which has standard header, legend and scale (which will preferably vary with the zone but could be fixed across all maps).  My first attempt was to get a new composer for each map -- that worked but was rather tedious.  Next I tried producing one map and printing it and then deleting the map frame and going into qgis to zoom to the next layer extent and make the boundary visible then hopping back into the composer to load it in.  This worked more or less.  Map was fine, legend was OK because it was the same for all the maps and the zone name was easy to edit but the scale needed to be deleted and set up again for each map.  Next I looked at Atlas Generation.  The manual seems to assume that you know how it works and all you need is a reminder about the details (like so many software manuals ;)  It seems to assume that you have something that you are iterating over but I can't see how to make it work -- there is nothing to iterate over to get the layers.  Any suggestions welcome!!  :)  Russell a two hour leave VS. two hours leave I couldn't get the difference between these?   give me a two hour leave  give me two hours leave    What is more, when it comes to using the latter, which one would you rather use? two hours' leave two hour leave