**Question 1:** For which rings $R$ does there exist a small set $S \subseteq Mod\_R$ such that every module $M \in Mod\_R$ is a direct sum of modules in $S$?


Equivalenty, for which rings $R$ does there exist a cardinal $\kappa$ such that every module $M \in Mod\_R$ is a direct sum of modules generated by $\leq \kappa$-many elements?


Faith [1] says that "$Mod\_R$ has a basis" if it answers to Question 1.


**Background:**


* Kothe showed that if $R$ is an artinian principal ideal ring, then every $R$-module is a direct sum of cyclic modules (i.e. $R$ answers to my question with $\kappa = 1$).


The general question with $\kappa=1$ seems to be generally known as "Kothe's problem".


* [Cohen and Kaplansky](https://doi.org/10.1007/BF01179851) showed that the converse holds if $R$ is commutative. [Warfield](https://www.doi.org/10.1007/BF01110928) extended this to show that if $R$ is commutative, then $R$ answers to my question if and only if $R$ is an artinian principal ideal ring.


So the question is only interesting when $R$ is noncommutative.


* [Nakayama](https://doi.org/10.3792/pia/1195579089) constructed a (noncommutative) ring $R$ such that every module is a direct sum of cyclic modules, and yet $R$ is not an artinian principal ideal ring.


So Warfield's theorem apparently doesn't extend in the most straightforward way to the noncommutative setting.


* [Faith and Walker](https://doi.org/10.1016/0021-8693(67)90035-X) showed that there exists a cardinal $\kappa$ such that every *injective* right $R$-module is a direct sum of $\leq \kappa$-generated modules if and only if $R$ is right Noetherian. Faith ([1] cf. also [Griffiths Thm 2.2](https://www.doi.org/10.1007/BF01350858)) then showed that if $R$ answers to Question 1, then $R$ is right Artinian.


So any ring answering to Question 1 is right Artinian, though not necessarily a principal ideal ring by Nakayama's result.


I happen to be interested in a special case:


**Question 2:** Same as Question 1, but assuming that $R$ is hereditary.


[1] Faith, "Big Decompositions of Modules" AMS Notices 18, Feb 1971 p. 400



Categories of modules generated under coproducts by a small set?
The rings satisfying your condition (for right modules) are the **right pure semisimple rings**. There are many equivalent conditions. You can find a lot of information in Section 4.5 of the book


*Prest, Mike*, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk). xxviii, 769 p. (2009). [ZBL1205.16002](https://zbmath.org/?q=an:1205.16002).


or you might find it easier to access the older paper


*Prest, Mike*, [**Rings of finite representation type and modules of finite Morley rank**](http://dx.doi.org/10.1016/0021-8693(84)90080-2), J. Algebra 88, 502-533 (1984). [ZBL0538.16025](https://zbmath.org/?q=an:0538.16025).


As you say, such a ring must be right artinian.


It is known that a ring is both left and right pure semisimple if and only if it has finite representation type (i.e., every module is a direct sum of indecomposable modules, and there are finitely many isomorphism types of indecomposable module), which is a left/right symmetric condition. And there is a longstanding conjecture about a strengthening of this.


**Pure Semisimplicity Conjecture:** A right pure semisimple ring has finite representation type.


Or equivalently this says that pure semisimplicity should be a left/right symmetric condition. There are many positive results for particular classes of rings.


In Question $2$ you say that you are particularly interested in the hereditary case. This doesn't make the conjecture easier, as Herzog proved that if there is a counterexample then there is a hereditary counterexample. Combining this with a result of Simson, it turns out that to prove the conjecture it would be enough to prove that a right pure semisimple hereditary ring is left artinian.


There is a lot of work on rings of finite representation type, especially hereditary ones. The fundamental result is [Gabriel's theorem](https://en.wikipedia.org/wiki/Gabriel%27s_theorem) classifying the finite dimensional hereditary algebras over an algebraically closed field with finite representation type as those Morita equivalent to path algebras of quivers whose underlying graph is a disjoint union of simply laced Dynkin diagrams. There are many generalizations; one for general hereditary artinian rings is


*Dowbor, P.; Ringel, Claus Michael; Simson, D.*, Hereditary Artinian rings of finite representation type, Representation theory II, Proc. 2nd int. Conf., Ottawa 1979, Lect. Notes Math. 832, 232-241 (1980). [ZBL0455.16013](https://zbmath.org/?q=an:0455.16013).


