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A question on the Bass-Papp theorem on injectives

Fri, 11/03/2017 - 04:33

Let $\{Q_{i}: i\in \mathcal{I}\}$ be a family of indecomposible injective modules over a commutative ring $R$ with identity. Is it true that the injective envelope of the direct sum of $Q_{i}$'s equals the direct product $\prod_{i\in \mathcal I}Q_{i}$? ( $R$ may not be Noetherian and $\mathcal{I}$ could be infinite).