Re: Darkstar Security CO.[ SHIPS WIP]
Posted: Sat May 25, 2013 5:46 pm
In mathematics, and more specifically in abstract algebra, a ring is an algebraic structure that abstracts and generalizes the basic arithmetic operations, specifically the operations of addition and multiplication. Rings are specifically studied in the branch of mathematics known as algebra, but are used in most areas of mathematics, including geometry and mathematical analysis. They allow mathematicians to apply theorems in elementary algebra to non-numerical objects like polynomials, series and functions. The formal definition of rings is relatively recent (end of 19th century), and is an example of the tendency of modern mathematics to introduce, study, and manipulate abstract structures.
Briefly, a ring is an abelian group with a second binary operation that is associative and is distributive over the abelian group operation. The abelian group operation is called "addition" and the second binary operation is called "multiplication" in analogy with the integers. One familiar example of a ring is the set of integers. The integers are a commutative ring, since a times b is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field (such as the real numbers) is a commutative ring in which one can do "division" by any nonzero element.
The ring theory consists of two main parts: the commutative ring theory, commonly known as commutative algebra, and the noncommutative ring theory. The former primarily concerns itself with problems and naturally occurring ideas in algebraic number theory and algebraic geometry. Important commutative rings include fields, polynomial rings, the coordinate rings of algebraic varieties and the rings of integers of number fields. On the other hand, the noncommutative theory is based on very different methods and, therefore, may not be viewed as a generalization of the commutative case. The major part of it is the structure theory: how a ring breaks up into simple pieces such as matrix rings. Many noncommutative rings come from analysis; e.g., operator algebras and rings of differential operators. Via group rings, the theory also found applications to the representation theory of groups. In geometry, the cohomology ring of a topological space constitutes an important geometric invariant of the space.
