Discrete Mathematical Structures by Pearson

7-42 Discrete Mathematical Structures

THEOREM 7.37 A ring R is without zero divisors if and only if cancellation laws hold in R,

that is, R is without zero divisors ⇔ cancellation laws hold in R.

Proof: Let R be a ring without zero divisors. Let a, b, and c be any three elements in R such that

a ≠ 0, and ab = ac

Now

ab = ac ⇒ a.(b - c) = 0

Since R is without zero divisors,

a.(b - c) = 0 and a ≠ 0 ⇒ b - c = 0

⇒ b = c

which shows that left cancellation law holds. Similarly, we can show that right cancellation law holds.

Conversely, suppose that right cancellation law holds in R.

If possible, let ab = 0, a ≠ 0 and b ≠ 0 (i.e., R has divisors of zero)

Now a ≠ 0, ab = 0 = a.0 ⇒ b = 0 (Left cancellation law)

This contradicts the assumption that b ≠ 0. ...