Question

Cancellation law may not hold in an arbitrary ring.

Answer 1

Let us show that cancellation law may not hold in an arbitrary ring. Let us consider the ring $(\mathbb Z_8,+,\cdot).$ Then $\overline{4}\cdot \overline{2}=\overline{4}\cdot \overline{6}=\overline{0}$, but $\overline{2}\ne\overline{6}.$ Therefore, cancellation law does not hold in this ring.