Proposition 16

Let A be a ring. The following properties are equivalent: (a) the set of maximal ideals of A is finite; (b) the quotient of A by its Jacobson radical is the direct composition of a finite number of fields.

Suppose that the quotient of A by its Jacobson radical $\mathfrak{R}$ is a direct composition of a finite number of fields. Then $A/\mathfrak{R}$ possesses only a finite number of ideals and a fortiori only a finite number of maximal ideals. As every maximal ideal contains $\mathfrak{R}$ (Algebra, Chapter VIII, § 6, no. 2, Definition 2), the maximal ideals of A are the inverse images of the maximal ideals of $A/\mathfrak{R}$ under the canonical homomorphism $A \to A/\mathfrak{R}$; hence they are finite in number.

Conversely, suppose that A has only a finite number of distinct maximal ideals $m_1, \ldots, m_n$. The $A/m_i$ are fields and it follows from § 1, no. 2, Proposition 5 that the canonical mapping $A \to \prod_{i=1}^n A/m_i$ is surjective; as its kernel $\bigcap_{i=1}^n m_i$ is the Jacobson radical $\mathfrak{R}$ (Algebra, Chapter VIII, § 6, no. 2, Definition 2), $A/\mathfrak{R}$ is isomorphic to $\prod_{i=1}^n A/m_i$.