Rings and Fields

A ring has two operations, $+$ and $\times$: it is an abelian group under $+$, its multiplication is associative and distributes over addition ($a(b+c) = ab + ac$). Examples: $\mathbb{Z}$, and $\mathbb{Z}_n$ for any $n$.

A field is a commutative ring with identity in which every nonzero element has a multiplicative inverse. So a field lets you add, subtract, multiply and divide (except by $0$). $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields; $\mathbb{Z}$ is only a ring (no integer inverse for $2$).

Worked example. $\mathbb{Z}_p$ is a field exactly when $p$ is prime. So $\mathbb{Z}_7$ is a field: every nonzero element has an inverse (e.g. $3^{-1} = 5$ since $3\times5=15\equiv1$). But $\mathbb{Z}_6$ is not a field: $2 \times 3 = 6 \equiv 0$, so $2$ and $3$ are zero divisors — nonzero elements whose product is $0$ — and zero divisors can never have inverses.