As the example above illustrates,
Theorem 16.1.18 is a modest beginning in the study of which algebraic manipulations are possible when working with rings. A fact in elementary algebra that is used frequently in problem solving is the cancellation law. We know that the cancellation laws are true under addition for any ring, based on group theory. Are the cancellation laws true under multiplication, where the group axioms canβt be counted on? More specifically, let
be a ring and let
with
When can we cancel the
βs in the equation
We can do so if
exists, but we cannot assume that
has a multiplicative inverse. The answer to this question is found with the following definition and the theorem that follows.
Now, here is why zero divisors are related to cancellation.
Hence, the only time that the cancellation laws hold in a ring is when there are no zero divisors. The commutative rings with unity in which the two conditions are true are given a special name.
Definition 16.1.23. Integral Domain.
A commutative ring with unity containing no zero divisors is called an integral domain.
In this chapter, Integral domains will be denoted generically by the letter
We state the following two useful facts without proof.
We close this section with the verification of an observation that was made in Chapter 11, namely that the product of two algebraic systems may not be an algebraic system of the same type.