Cohen and Kaplansky have proven that a commutative ring $R$ has the property



> 
> C: Every $R$-module is a direct sum of cyclic $R$-modules.
> 
> 
> 


if and only if $R$ is an Artinian principal ideal ring. Can we classify those commutative rings $R$ with the following property?



> 
> C': Every $R$-module is a direct summand of a direct sum of cyclic $R$-modules.
> 
> 
> 


Actually I am interested in the following property, which is more involved:



> 
> C'': For every $R$-module there is a commutative $R$-algebra $R'$ satisfying $[\forall T \in \mathsf{Mod}(R):~R' \otimes\_R T = 0 \Rightarrow T=0]$ (for example, a faithfully flat algebra) such that the $R'$-module $R' \otimes\_R M$ is a direct summand of a direct sum of localizations of cyclic $R'$-modules.
> 
> 
> 


If $R$ is arbitrary, what can we say about this class of $R$-modules?



The closure of cyclic modules under direct sums and direct summands
The introduction of


*Warfield, R.B.jun*, [**Rings whose modules have nice decompositions**](http://dx.doi.org/10.1007/BF01110928), Math. Z. 125, 187-192 (1972). [ZBL0218.13012](https://zbmath.org/?q=an:0218.13012).


says



> 
> It is also shown that if $R$ is a commutative ring and there is a cardinal number $\mathfrak{n}$ such that every $R$-module is a summand of a direct sum of modules with at most $\mathfrak{n}$ generators, then $R$ is an Artinian principal ideal ring.
> 
> 
> 


That seems to answer question C'.


