It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective if and only if every injective $R$-module is a direct sum of indecomposable injective $R$-modules.

1) Is it true that a Grothendieck category $\mathcal{C}$ is locally noetherian if and only if every injective object is a direct sum of indecomposable injective objects?

2) Is it true that a scheme $X$ in algebraic geometry is (locally) noetherian if and only if every injective quasi-coherent sheaf is a direct sum of indecomposable injective quasi-coherent sheaves?