AN SPECIAL GROUP !

Hi everyone again !! Today we are going to discuss about an special group ---> The Abelian Group. 

Let's say that an abelian group (G,+)  is a set G paired with a commutative binary operation +, where G has a special identity element called 0r which acts as an identity for +. The typical example of an abelian group is, the integers Z with the addition as a condition, with zero playing the role of the null element.

The easiest way to think of a ring is as an abelian group with more structure. This structure comes in the form of a multiplication operation which is “compatible” with the addition coming from the group structure.

A ring (R,+, · ) is a set R which forms an abelian group under + (with neutral element --> 0), and has an additional operation with an element 1 as a neutral element . Furthermore,  distributes over + in the sense that for all 

 and 

The most important thiing to note is that multiplication is not comutative both in general rings and for most rings in practize. If multiplication is comutative, then the ring is called commutative. Some easy examples of commutative rings include rings of numbers like Z,Q,R, which are just the abelian groups that we already know.