A division ring (skew field) is a ring where every non-zero element has a multiplicative inverse. $(\mathbb{Q},+,\cdot)$ is a field (and hence a division ring): every non-zero rational $p/q$ has inverse $q/p\in\mathbb{Q}$. $\mathbb{Z}$ has no inverses for most elements; $\mathbb{Z}_6$ has zero divisors ($2\cdot 3=0$); $E$ (even integers) lacks unity.