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 .
Definition 1 – 3
A ring with identity is a ring in which ? an identity element
for the multiplication operation (it is denoted by 1 ).
Definition 1 – 4
In the ring R with identity , we say that a is invertible ( unit )
if ? invers of a with respect to multiplication where a ? R ,
we denoted it by a-1 .
Notes
1 -The invers of any element is unique if it exist .
2-The set of all invertible elements in a ring R is denoted by R*.
Remark
( R* , . ) is group , it is called group of invertible elements .
Proof
H . W .
Exercises
1 - Find the groups of invertible elements of:
( R , + , . ) , ( Z4 , +4 , .4 ) , (Z5 , +5 , .5 ) , and (Z6 , +6 , .6 )
2 - Show that wither each of the following is a ring or not :
a – ( P (X) , U , ?)
b - ( P (X) , ? , ? )
c – ( Mn [R] , + , . )