Direct sum of injective modules is injective.

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an arbitrary abelian subcategory $\mathcal{C}$ of the category $R\text{-mod}$ of unitary left $R$-modules.

We say that an abelian subcategory $\mathcal{C}$ of $R\text{-mod}$ is injectively closed if it satisfies the property that, for an arbitrary family $\left(I_\alpha\right)_{\alpha\in A}$ of injective objects in $\mathcal{C}$ such that $I:=\bigoplus\limits_{\alpha \in A}\,I_\alpha$ is an object in $\mathcal{C}$, $I$ is an injective object in $\mathcal{C}$. Is it true that if $R$ is left Noetherian, then any abelian subcategory of $R\text{-mod}$ is injectively closed? If not, can you please provide a counterexample? Is there a sufficient condition for $\mathcal{C}$ to be injectively closed? References are greatly appreciated.

For a nontrivial example, let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic $0$ with a triangular decomposition $$\mathfrak{g}=\mathfrak{n}^-\oplus \mathfrak{h}\oplus \mathfrak{n}^+\,.$$ Denote by $\bar{\mathcal{O}}$ the full subcategory of the category of $\mathfrak{U}(\mathfrak{g})$-modules (where $\mathfrak{U}(\mathfrak{g})$ is the enveloping algebra of $\mathfrak{g}$) consisting of $\mathfrak{U}(\mathfrak{g})$-modules $M$ with the following properties:

1. $M$ is a weight module with respect to the Cartan subalgebra $\mathfrak{h}$,

2. each weight space of $M$ is finite dimensional, and

3. $M$ is locally $\mathfrak{n}^+$-finite (that is, $\mathfrak{U}\left(\mathfrak{n}^+\right)\cdot v$ is a finite-dimensional vector subspace of $M$ for any $v\in M$).

Then, $\bar{\mathcal{O}}$ is injectively closed. (In this example, note that $\mathfrak{U}(\mathfrak{g})$ is both left and right Noetherian.)

P.S.: The Bass-Papp Theorem can be found, for example, in Theorem 3.39 on Page 123 of An Introduction to Homological Algebra by Joseph Rotman.

• Do you want to assume your subcategory is full? Otherwise, for instance, any category of modules is equivalent to an abelian subcategory of $Ab$. May 11, 2017 at 2:03
• @EricWofsey No, I don't assume that, but this assumption doesn't hurt. I will only consider full subcategories in whatever I am dealing with anyway. By the way, could you please give me a reference to the statement that any category of modules is equivalent to an abelian subcategory of $\text{Ab}$? May 11, 2017 at 9:43
• Just take a skeleton of $RMod$ such that no two objects are equal as abelian groups and then take the forgetful functor to $Ab$. May 11, 2017 at 13:51