Give a one-sentence synopsis of the proof of the "if" part of Theorem 19.5. 

Data from 19.5 Theorem 

The cancellation laws hold in a ring R if and only if R has no divisors of 0. 

Proof:  Let R be a ring in which the cancellation laws hold, and suppose ab = 0 for some a, b ∈ R. We must show that either a or b is 0. If a ≠ 0, then ab= a0 implies that b = 0 by cancellation laws. Similarly, b ≠ 0 implies that a = 0, so there can be no divisors of 0 if the cancellation laws hold. 

Conversely, suppose that R has no divisors of 0, and suppose that ab = ac with a ≠ 0. Then 

ab - ac = a(b - c) = 0. 

Since a ≠ 0, and since R has no divisors of 0, we must have b - c = 0, sob = c. A similar argument shows that ba = ca with a ≠ 0 implies b = c.