ZERO DIVISORS
Definition 1 – 9
If R is any ring and 0 ? a ? R , then a is called a zero
divisor in R if ? b ? 0 ? ab = 0 .
Example
In ( Z6 ,+6 , .6 ) , [ 3 ] .6 [4 ] = [ 0 ] , then [ 3 ] , [ 4 ] are zero divisors
of Z6 .
Example
If X = { a , b } , then P ( X ) = { { a } , { b } , X , ? } .
{ a } ? { b } = ? , in which {a} , {b} ?? , then {a} , {b} are zero divisors
of ( P(X) , ? , ? ) .
Remark
In a ring R , the cancellation law is not necessary to be hold .
Example
[3] .6 [2] = [3] .6 [4] , but [2] ? [4] .
Theorem 1 – 10
A ring R is with out zero divisors iff it satisfies the cancellation
laws for multiplication .
proof .
Suppose that R is without zero divisors and let ab = ac , a? 0 ,
ab – ac = 0 , so a( b – c ) = 0 .
Hence b- c =0 ? , then b = c .
Similarly , if ba = ca , then b = c .
Conversely , let R satisfy the cancellation laws and assume that
ab = 0 , with a ? 0 , then ab = a0.
Hence b = 0 ?
Similarly , b ? 0 implies a = 0 .
Hence R have no zero divisors .