الحلقات

RINGS

Last year we studied groups , in this year we will study rings
and some concepts which jointing with rings .

Definition 1 - 1
Let R be a nonempty set with two binary operations * and o .
(R , * , o ) is called a ring iff
( i) ( R , * ) is a commutative group .
( ii) (R , o ) is a semigroup .
( iii) o is distributive over * ; that is, ? a , b , c ? R .
a o ( b * c ) = a o b * a o c, (a * b ) o c = a o c * b o c .

Examples :
(Z , + , . ) , ( Q , + , . ) , ( R , + , . ) , ( C , + , . ) , and ( Zn , +n , .n ) are
rings .

Notes
1- * and o are called addition and multiplication respectively , we will use + and . instead of * and o respectively .
2- The identity of the additive commutative group is called zero
element of the ring R and it is unique . We denoted the zero
element of a ring by 0 .
3- The additive inverse of an element a of ( R , + ) is denoted by - a . Thus , in a ring R , 1
Definition 1 – 2
A commutative ring ( R , + , . ) in which multiplication is
a commutative operation
a . b = b . a ? a , b ? R .