Question

22. (a) If ab is a zero divisor in a ring R, prove that a or b is a zero divisor. (b) If a or b is a zero divisor in a commutative ring R and ab ? 0R, prove that ab is a zero divisor.

          22. (a) If ab is a zero divisor in a ring R, prove that a or b is a zero divisor.
(b) If a or b is a zero divisor in a commutative ring R and ab ? 0R, prove that
ab is a zero divisor.

Added by -Scar C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/11/2023

Step 1

This means there exists a nonzero element $c \in R$ such that $(ab)c = 0$. Since $R$ is a ring, we have the associative property, so we can rewrite this as $a(bc) = 0$. Now, either $a$ or $bc$ must be a zero divisor. If $a$ is not a zero divisor, then it must Show more…

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22. (a) If ab is a zero divisor in a ring R, prove that a or b is a zero divisor. (b) If a or b is a zero divisor in a commutative ring R and ab ≠ 0R, prove that ab is a zero divisor.