Corollary 1

If the ring $A$ is left Noetherian, every direct sum of injective $A$-modules is injective.

Let $(E_i)_{i \in I}$ be a family of injective $A$-modules, let $E$ be their direct sum, let $\alpha$ be an ideal of $A$ and let $u : \alpha \to E$ be an $A$-homomorphism. Since $A$ is noetherian, $\alpha$ is of finite type, and consequently the canonical mapping

is bijective; let $(u_i)$ be the inverse image of $u$ by $\varphi$. Since each $E_i$ is injective, and the family $(u_i)$ has finite support, there exists an element $(e_i)_{i \in I}$ of $E$ such that $u_i(a) = ae_i$ for all $a \in \alpha$ and all $i \in I$, hence $u(a) = a((e_i))$ for all $a \in \alpha$, and $E$ is injective.