A Study on Algebra of Groups and Rings Structures in Mathematics

Abstract

Algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Algebra deals with the more general concept of sets is a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Set theory is a branch of logic and not technically a branch of algebra. Binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (−), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a, and is necessarily unique, if it exists. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element. The inverse of a is written −a, and for multiplication the inverse is written a −1 . A general two-sided inverse element a −1 satisfies the property that a ∗ a −1 = e and a −1 ∗ a = e, where e is the identity element. Associativity is, the grouping of the numbers to be added does not affect the sum is (2 + 3) + 4 = 2 + (3 + 4). Commutative is, the order of the numbers does not affect the result is 2 + 3 = 3 + 2. Combining the concepts gives group and ring one of the most important structures in mathematics. A group is a combination of a set S and a single binary operation is an identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a. A group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—then the group is said to be abelian. A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a. Keyword: Groups, Rings and Fields are axiomatically and algebra