mysticmanchot's New Writeupshttp://everything2.com/?node=New%20Writeups%20Atom%20Feed&foruser=mysticmanchot2004-08-30T07:33:02Zgroup ring (thing)http://m.everything2.com/user/mysticmanchot/writeups/group+ringmysticmanchothttp://m.everything2.com/user/mysticmanchot2004-08-30T07:33:02Z2004-08-30T07:33:02Z<p>
A Group Ring is, as the name suggests, a combination of two simpler algebraic objects, namely <a href="/title/groups">groups</a> and <a href="/title/rings">rings</a>. A group ring may be thought of as a sort of <a href="/title/direct+product">direct product</a> between a <a href="/title/group">group</a> and a <a href="/title/ring">ring</a>. As this might suggest a the group ring RG (R a <a href="/title/ring">ring</a> G a <a href="/title/group">group</a> consists of all possible <a href="/title/linear+combination">linear combination</a>s of the elements of G over the ring R, and the new hybrid RG is itself a ring (which contains a copy of G in the <a href="/title/unit+group">unit group</a> and a copy of R in the ring itself).
</p><p>
For example if our group G is the group {e,a,b} (The Cyclic group of order 3 with identity e) an element of ZG is 1*e+75*a+33*b. Another example would be 0*e+4*a+2*b, but when we have 0*x we usually choose to omit that term, so the latter may also be written as 4*a+2*b. Also to make the <a href="/title/embedding">embedding</a> of R clear we traditionally omit the '*e' on the term which is multiplied by the identity, so that if r is an element of r then we may write that r is an element of RG, and we will mean r= r*e+0*g1+0*g2+... and…Unit group (thing)http://m.everything2.com/user/mysticmanchot/writeups/Unit+groupmysticmanchothttp://m.everything2.com/user/mysticmanchot2004-08-30T02:12:08Z2004-08-30T02:12:08Z<p>
The Unit Group of a <a href="/title/Unit+Ring">Unit Ring</a> (the mathematical object, not your ordinary household or circus variety) is the set of all elements of a ring which are <a href="/title/invertible">invertible</a> (ie those elements which may be multiplied by some other element to get 1). For example note the element 1 is in the unit group of every unit ring because 1 is itself invertible. It is possible that 1 is the only invertible element in a <a href="/title/ring">ring</a> (ie in the ring of integers mod 2).
</p><p>
The Unit Group is called such because the set of elements that make up the unit group of a <a href="/title/ring">ring</a> form a <a href="/title/group">group</a>.
</p><p>
Not all groups can occur as the unit group of a <a href="/title/ring">ring</a>, the smallest example of a group which is never the unit group of a ring is the cyclic group of order <a href="/title/5">5</a> (proving this amounts to a nice exercise in algebra).
</p><p>
Determining which groups occur as the unit groups of rings is a nontrivial problem which has been worked on by Thomas Occhipinti.
</p><p>
A current conjecture is that the only <a href="/title/prime">prime</a> order groups which…Mississippi School for Mathematics and Science (place)http://m.everything2.com/user/mysticmanchot/writeups/Mississippi+School+for+Mathematics+and+Sciencemysticmanchothttp://m.everything2.com/user/mysticmanchot2004-08-30T01:36:35Z2004-08-30T01:36:35Z<p>
The Mississippi School for Mathematics and Science (or MSMS as it is called for short) is a residential, public magnet high school located in Columbus, <a href="/title/Mississippi">Mississippi</a>. It is for Juniors and Seniors exclusively, and has graduated 16 classes of students. The School is beloved by the students, and is a passion for most of the faculty who could all be paid much better to do other things if they were not devoted to the cause (indeed 50% of the faculty hold PhDs). In short MSMS is the ideal paradise for the Open Minded and the Academic Oriented students of Mississippi.
</p><p>
Each MSMS class has about 120 students, who live in two dorms, Fant for the girls and Peyton for the guys. MSMS's budget constitutes about one tenth of Mississippi's whole education budget, and the school estimates a cost of about 15,000 dollars for each student that goes through the school. The school itself is located on the Mississippi University for Women campus, about 30 minutes east of Starkville.</p>
Tommy, class of 2001