Definition 4

The ideal of nilpotent elements of a (commutative) ring $A$ is called the nilradical of $A$. If $a$ is an ideal of $A$, the inverse image, under the canonical mapping $A \to A/a$, of the nilradical of $A/a$ is called the radical of $a$.

We often denote by $r(a)$ the radical of an ideal $a$ of $A$.

To say that an element $x \in A$ belongs to the radical of $a$ means therefore that there exists an integer $n > 0$ such that $x^n \in a$. Iff is a homomorphism from $A$ to a ring $B$ and $b$ is an ideal of $B$, the radical off $^{-1} (b)$ is the inverse image underf of the radical of $b$, for to say that $x^n \in f^{-1} (b)$ means that $(f(x))^n \in b$.

The nilradical of a ring $A$ is contained in its Jacobson radical (Algebra, Chapter VIII, § 6, no. 3, Corollary 3 to Theorem 1) but may be distinct from it; it is always equal to it if $A$ is Artinian (Algebra, Chapter VIII, § 6, no. 4, Theorem 3).

We say that a prime ideal $p$ of a ring $A$ is a minimal prime ideal if it is minimal in the set of prime ideals of $A$ ordered by inclusion.