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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 showed that the converse holds if $R$ is commutative. Warfield 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 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 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) 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

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    $\begingroup$ For some reasons a sort of dual question is also interesting: For which rings $R$ does there exist a small set of modules (equivalently a single module) $S$ such that every module is direct summand (=direct factor=retract) of a product of modules in $S$? (see mathoverflow.net/questions/68352/… ) $\endgroup$ – George C. Modoi May 29 '20 at 14:08
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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.

or you might find it easier to access the older paper

Prest, Mike, Rings of finite representation type and modules of finite Morley rank, J. Algebra 88, 502-533 (1984). ZBL0538.16025.

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 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.

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    $\begingroup$ Thanks so much -- I hadn't dared hope for such a complete answer! It's interesting that the pure semisimplicity conjecture would imply that if $Mod_R$ has a basis, then it has a finite basis of indecomposable modules. This would be surprising, but definitely in the spirit of Warfield's theorem, which says that if $R$ is commutative and $Mod_R$ has a basis, then it has a basis of cyclic modules. $\endgroup$ – Tim Campion May 10 '20 at 13:02

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