Expert Answer to "Show that a commutative ring with cancellation property..."

High School

Answered question

2022-05-03

Show that a commutative ring with cancellation property has no zero divisor

Answer & Explanation

karton

Expert2022-07-07Added 613 answers

Let $a, b, c \in R$ , and $a \neq 0$ . The cancellation property states that $a b = a c$ implies $b = c$ . We want to show that if the cancellation property holds, then there are no zero divisors.

Suppose the opposite, that there do exist zero divisors. Then for zero divisor $a \neq 0$ there exists $b \neq 0$ so that $a b = 0$ . This means that $a b = 0 = a 0$ . If cancellation holds, then $b = 0$ . But we assumed that $b \neq 0$ , which is a contradiction. Therefore, cancellation cannot hold.

Hence, cancellation implies no zero divisors.

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