Page 188 - DMTH403_ABSTRACT_ALGEBRA

Unit 18: Integral Domains

Definition: A non-zero element a in a ring R is called a zero divisor in R if there exists: a Notes
non-zero element b in R such that ab = 0.

Now do you agree that 2 is a zero divisor in Z,? What about 3 in Z ? Since 3 x  0 for every
4
non-zero x in Z , 3 is not a zero divisor in Z .
4
4
Now let us look at an example of a zero divisor in C[0, l]. Consider the function
f  C[0, 1] given by

 1 x 1/2
  x  2 , 0  
f(x)   x 1
 0,1/2  
 

Let us define g : [0, 1] + R by

x
 0, 0   1/2
g(x)   x 1

 x 1/2,1/2  
Then g  C[0, 1], g  0 and (fg) (x) = 0  x  [0,1]. Thus, fg is the zero function. Hence, f is a zero
divisor in C[0, 1].
For another example, consider the Cartesian product of two non-trivial rings A and B. For every
a  0 in A, (a, 0) is a zero divisor in A × B. This is because, for any b  0 in B. (a . 0) (0.b) = (0.0).
Now let us look at the ring (X), where X is a set with at least two elements, Each non-empty
proper subset A of X is a zero divisor because A.X = AA = , the zero element of (X).
C
C
Let us now talk of a type of ring that is without zero divisors.
Definition: We call a non-zero ring R an integral domain if
(i) R is with identity, and

(ii) R has no zero divisors.
Thus, an integral domain is a non-zero ring wilh identity in which the product of two non-zero
elements is a non-zero element.

This kind of ring gets its name from the set of integers, one of its best known examples. Other
examples of domains that immediately come to mind are Q, R and C. What about C[0,1]? You
have already seen that it has zero divisors. Thus C[0,l] is not a domain.

Note Several authors often shorten the term ‘integral domain’ to ‘domain’. We will
do so too.
The next result gives us an important class of examples of integral domains.

Theorem 1: Z is an integral domain iff p is a prime number.
p
Proof: Firstly, let let us assume that p is a prime number. Then you know that Zp is a non-zero
ring with identity. Let us see if it has zero divisors. For this, suppose a,b Z p satisfy a,b  0.

Then ab  0, i.e., p | ab. Since p is a prime number, we see that p | a or p | b. Thus, a 0 or b 0.

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