Groups, Rings, and Fields
As cluster algebras were stated earlier to be “rings,” we should spend some time getting to know just what rings are --- but, to understand rings, we must first introduce the concept of groups. A group is simply a set of objects that obey the following properties:
1.) There exists a binary composition rule, such that for any elements x and y, there can be a composition x ￮ y. The composition of any two elements must be another element of the group.
2.) Composition obeys the associative property (much like that of addition among integers). However, commutativity is not required.
3.) There exists an identity element e, such that x ￮ e = x and e ￮ x = x.
4.) There exists an inverse element x-1 for every element x, such that x ￮ x-1 = e.
For example, the set of integers is a group because it is closed under addition, addition is associative, the identity element is “0,” and there is always an additive inverse for an integer.
Rings are like groups (they obey every property of a group for their first binary composition, noted as “+”), but have an additional binary operation often referred to as “multiplication.” They obey an additional set of rules, outlined below:
1.) The underlying group structure of the ring must be commutative; this means that x + y = y + x for all x and y in the ring.
2.) For any elements x and y, the product xy must be a member of the ring.
3.) A distributive property exists, such that x(y + z) = xy + xz = (y + z)x.
4.) The multiplication operation is associative (just like that of integer multiplication).
5.) There is a multiplicative identity element, “1,” such that 1·x = x·1 for all x in the ring.
6.) The product of an element with the additive inverse (denoted by “0” here instead of e) is equal to 0.
Note: the existence of a multiplicative inverse is not required.
For example, the set of integers is an infinite ring with addition and multiplication, is it follows the rules above exactly.
As a final note, a “field” is simply a commutative ring (multiplication is commutative) such that it is not the zero-ring (it has more than one element) and every element has a multiplicative inverse. For example, the set of integers is not a field because it does not contain an element x such that 3x = 1. However, the set of rational functions is a field because it obeys every property of a commutative ring and contains multiplicative inverses for each element --- for example, x2 is the multiplicative inverse of x-2.
