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).
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1$\begingroup$ No, see mathoverflow.net/a/53935. $\endgroup$– Fred RohrerCommented Nov 3, 2017 at 12:13
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1$\begingroup$ Surely no because $R$ may be Noetherian and $I$ may be infinite whence the injective envelope of the direct sum of the $Q_i$ is the direct sum of the $Q_i$ which is not the product of the $Q_i$. Or did you mean to insist that $R$ be non-Noetherian. $\endgroup$– Simon WadsleyCommented Nov 3, 2017 at 17:20
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