Exercise 22, § 9

Let A be a commutative integral domain that has only a finite number $s >1$ of maximal ideals $\mathfrak{m}_1, . . . ,\mathfrak{m}_s$.

a) Prove that the ring $A/\mathfrak{R}(A)$ is isomorphic to the product of the fields $A/\mathfrak{m}_i$.

b) Prove that the A-modules $A/\mathfrak{m}_i$ do not admit a projective cover even though they are projective and finitely generated as $A/\mathfrak{R}$(A)-modules (observe that A is a projective cover of $A/\mathfrak{R}(A)$).

c) Let K be a commutative field. Prove that the subring A of the field K(T) consisting of the rational fractions that can be written as $P(T)/Q(T)$ with $Q(0)\not= 0$ and $Q(1)\not= 0$ has the required properties.