Integral domains

$\mathbb{Z}$. Commutative with unit, and $ab=0\Rightarrow a=0$ or $b=0$ (for integers). Integral domain.

$\mathbb{R}[X]$. Commutative with unit, and the product of two nonzero polynomials is nonzero (degree adds). Integral domain.

$\mathbb{Z}/p\mathbb{Z}$. For $p$ prime, if $\bar a\bar b=\bar0$, then $p\mid ab$, so $p\mid a$ or $p\mid b$ (Euclid's lemma). Hence $\bar a=\bar0$ or $\bar b=\bar0$. Integral domain.

$\mathbb{Z}/6\mathbb{Z}$. $\bar2\cdot\bar3=\bar0$ with $\bar2,\bar3\neq\bar0$. Not an integral domain.

Conclusion. The cancellation law holds in integral domains: $ab=ac$ and $a\neq0$ imply $b=c$.