Hw12 Problem 9

Return to Homework 12, Glossary, Theorems

Problem: Show by example that for fixed nonzero elements $a$ and $b$ in a ring, the equation $ax=b$ can have more than one solution. How does this compare with groups?

Solution: Let $a = 6$ and $b = 0$ in $\mathbb{Z}_{12}$. We can see that $x = 2, 4, 6, 8,$ and $10$ are all solutions to this equation. Groups are different because the cancellation laws always hold. For rings, cancellation laws only hold when there are no divisors of 0, as stated by Theorem 19.5.

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