Does $ab = b^2 \implies a=b? $

Question

My textbook says that the follow is not true: $ab = b^2 \implies a=b $

However, I cannot find a single case where its not true.

What am I missing? Is it true or false? How do I go about proving these things?

Answer 1

$5 * 0 = 0 * 0$ but $5 \neq 0$.

In general, you need cancellation for that property to hold. Cancellation is a property that allows one to derive the fact that

$$ ab = ac \implies b = c $$

One example is $\mathbb Z - \{0\}$.

Answer 2

Try $b=0$. Then $a\in\Bbb{R}\ni\{0\}$. So $a=0$ is not the unique solution and surely, $a$ not necessarily equals to $b$.

Answer 3

Consider $a=1$ and $b=0$. We see $ab = (1)(0) = 0 = 0^2 = b^2$ but $a \neq b$.

Answer 4

i would write $$ab=b^2$$ and this is equivalent to $$0=b(a-b)$$ thus we get $$a=b$$ or $$b=0$$