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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?

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  • $\begingroup$ What is the meaning of "the $R$-module $M$ is a localization of a cyclic module"? Is it "there exists $S\subset R$ multiplicative and a cyclic $R$-module $N$ such that $M$ is isomorphic to $S^{-1}N$ as $R$-module (where $S^{-1}N=N\otimes_R S^{-1}R$) $\endgroup$ – YCor Apr 22 '17 at 8:15
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    $\begingroup$ @YCor. Yes sure. (What else could it mean?) $\endgroup$ – Martin Brandenburg Apr 22 '17 at 8:15
  • $\begingroup$ I couldn't think of another, but I wasn't convinced there's none... $\endgroup$ – YCor Apr 22 '17 at 8:35
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The introduction of

Warfield, R.B.jun, Rings whose modules have nice decompositions, Math. Z. 125, 187-192 (1972). ZBL0218.13012.

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'.

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