Page 189 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes
What we have shown is that if a  0 and b  6, then ab  6. Thus, Z is without zero divisors, and
p
hence, is domain.
Conversely, we will show that if p is not a prime, then Z is not a domain, So, suppose p is not
p
a prime. If p = 1, then Z , is the trivial ring, which is not a domain.
If p is composite number and m | p, you know that m  Z is a zero divisor. Thus, Z has zero
p
p
divisors. Hence, it is not a domain.

Task Which of the following rings are not domains? Why?
Z , Z , 2Z, Z + iZ, R × R, {0}.
4
5
Now consider a ring R. We know that the cancellation law for addition holds in R, i.e., whenever
acb = acc in R, then b = c. But, does ab = ac imply b = c? It need not. For example, 0.1 = 0.2 in Z but
1 # 2. So, if a = 0, ab = ac need not imply b = c. But, if a # 0 and ab = ac, is it true that b = c’? We will
prove that this is true for integral domains.
Theorem 2: A ring R has no zero divisors if and only if the cancellation law for multiplication
holds in R (i.e., if a, b, c  R such that a  0 and ab = ac, then b = c.)
Proof: Let us first assume that R contains no zero divisors. Assume that a, b, c  R such that
a  0 and ab = ac. Then a(b – c) = ab – ac = 0. As a  0, and R has no zero divisors, we get b – c = 0,
i.e., b = c.
Thus if ab = ad and a  0, then b = c.
Conversely, assume that the cancellation law for multiplication holds in R. Let a  R such that
a  0. Suppose ab = 0 for some b  R. Then ab = 0 = a0. Using the cancellation law for multiplication,
we get b = 0. So, a is not a zero divisor, i.e., R has no zero divisors.
Using this theorem we can immediately say that the cancellation law holds for multiplication in
an integral domain.

Now let us introduce a number associated with an integral domain in fact, with any ring.
x
0
For this let us look at Z first. We know that 4x    Z . In fact, 8x = 0 and 12 x = 0 also for
4
4
any x  Z .
4
But 4 is the least element of the set { n  N | nx = 0  x  Z ). This shows that 4 is the
4
characteristic of Z , as you will see now.
4
Definition: Let R be a ring. The least positive integer n such that nx = 0  x  R is called the
characteristic of R. If there is no positive integer n such that nx = 0  x  R, then we say that the
characteristic of R is zero.
We denote the characteristic of the ring R by char R.
You can see that char Z = n and char Z = 0.
n
Now let us look at a nice result for integral domains. It helps in considerably reducing our
labour when we want to obtain the characteristic of a domain.

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