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. 



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> 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.
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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. I:** The Bass-Papp Theorem can be found, for example, in Theorem 3.39 on Page 123 of [An Introduction to Homological Algebra](https://books.google.de/books?id=P2HV4f8gyCgC&printsec=frontcover&dq=An%20Introduction%20to%20Homological%20Algebra%20By%20Joseph%20Rotman&hl=en&sa=X&ved=0ahUKEwjZidmbrObTAhUBPVAKHZ-9CfkQ6AEIIzAA#v=onepage&q=An%20Introduction%20to%20Homological%20Algebra%20By%20Joseph%20Rotman&f=false) by Joseph Rotman.


**P.S. II:** I posted this question [here](https://math.stackexchange.com/questions/2275475/direct-sum-of-injective-modules-is-injective) as well, but didn't receive any answer. I figured that MathOverflow users may be able to help. I don't know how to move my Math.StackExchange post to MathOverflow.



Direct sum of injective modules is injective
**EDIT** Thanks to Jeremy Rickard for several corrective insights in the comments!


In the direction of positive conditions, I'm not sure whether the following conditions are reasonable for your purposes:


**Proposition:** Let $\mathcal{C}$ be an abelian category which is 


* [locally finitely presentable](https://ncatlab.org/nlab/show/locally+finitely+presentable+category) (i.e. it has a generator of [finitely presentable objects](https://ncatlab.org/nlab/show/compact+object) and is complete, or equivalently cocomplete) and
* "strongly Noetherian" in the sense that any subobject of a finitely-presentable object is finitely-presentable, and similarly for higher presentability degrees.


Then $\mathcal{C}$ is injectively closed.


**Proof sketch:** One shows that in such a category, an object is injective if and only if it lifts against monomorphisms between finitely-presentable objects. This uses the "strong Noetherian" property. This implies, in conjunction with the Noetherianity property again, that injective objects are closed under filtered colimits; since they are also closed under finite direct sums, they are thus closed under arbitrary direct sums.


**Corollary:** Let $R$ be a "strongly Noetherian ring" in the sense that $R$-$\mathrm{Mod}$ is "strongly Noetherian", and suppose that $\mathcal{C}$ is a full abelian subcategory of $R$-$\mathrm{Mod}$ which is 


* closed under kernels and colimits in $R$-$\mathrm{Mod}$, and
* generated under colimits by finitely-presentable $R$-modules.


Then $\mathcal{C}$ is injectively closed.


**Proof sketch:** Check that $\mathcal{C}$ satisfies the hypotheses of the proposition.


Notes:


1. As Jeremy Rickard points out, for countable $R$, $R$-$\mathrm{Mod}$ is "strongly Noetherian" iff it is Noetherian. So it seems that it is not that much stronger a condition than simply being Noetherian.
2. In Jeremy Rickard's example, where $\mathcal{C}$ is the subcategory of $R$-$\mathrm{Mod}$ given by the image of $S$-$\mathrm{Mod}$, it is the case that $\mathcal{C}$ is closed under kernels and colimits, and is itself locally finitely presentable. But it doesn't satisfy the hypotheses of the corollary because $\mathcal{C}$ is not generated under colimits by modules which are finitely-presentable *as $R$-modules*. Another way to say this is that the embedding $S$-$\mathrm{Mod} \to R$-$\mathrm{Mod}$ doesn't preserve finite presentability: the module $S$ itself is finitely-presentable as an $S$-module but not as an $R$-module.


