Exercise 12, § 2

(a) For a product $\prod_{\lambda \in L} E_\lambda$ of $A$-modules to be injective (Exercise 11), it is necessary and sufficient that each of the $E_\lambda$ be injective. (b) Let $K$ be a commutative field, $L$ an infinite set, $A$ the product ring $K^L$. Now prove that $A_s$ is an injective $A$-module (use (a)), but that the ideal $a$ of $A$, direct sum of the factors of $K^L$ (canonically identified with ideals of $A$), is not an injective $A$-module.