Exercise 10, § 9

Let $A$ be a commutative ring and $\mathscr{U}$ the intersection of its maximal ideals.

(a) Show that the canonical homomorphism $A^* \to (A/\mathscr{U})^*$ is surjective and that its kernel is $1 + \mathscr{U}$.

(b) Suppose that the set $M$ of maximal ideals of $A$ is finite and that $A^*$ is finite. Deduce that $A$ is finite (apply Proposition 8 of § 8, no. 10 to $A/\mathscr{U}$).

(c) Deduce that the set of prime numbers is infinite (note that $\mathbf{Z}^* = \{1, -1\}$).

(d) Suppose that $A^*$ is finite and that $A/m$ is finite for every maximal ideal $m$. Can it be deduced that $A$ is finite, that $A$ is countable?