A commutative ring $R$ is an integral domain if $ab = 0$ implies $a = 0$ or $b = 0$ (no zero divisors). A ring $R$ is a field if every non-zero element has a multiplicative inverse.

Every field is an integral domain, but not conversely. The integers $\mathbb{Z}$ form an integral domain that is not a field.

An ideal $\mathfrak{p} \subsetneq R$ is prime if whenever $ab \in \mathfrak{p}$, we have $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$. An ideal $\mathfrak{m} \subsetneq R$ is maximal if there is no ideal $I$ with $\mathfrak{m} \subsetneq I \subsetneq R$.

Key characterizations:

• $\mathfrak{p}$ is prime $\Leftrightarrow$ $R/\mathfrak{p}$ is an integral domain
• $\mathfrak{m}$ is maximal $\Leftrightarrow$ $R/\mathfrak{m}$ is a field

The radical of an ideal $I$ is: $\sqrt{I} = \{a \in R : a^n \in I \text{ for some } n \geq 1\}$

An ideal is radical if $I = \sqrt{I}$. Prime ideals are always radical. The radical corresponds geometrically to removing multiplicity from geometric objects.

An element $a \in R$ is nilpotent if $a^n = 0$ for some $n \geq 1$. The set of all nilpotent elements forms an ideal called the nilradical, equal to $\sqrt{(0)}$. An element $u \in R$ is a unit if there exists $v \in R$ with $uv = 1$.