Let $R$ be a commutative ring with unity. The Bass-Papp theorem states that any countable direct sum of injective $R$-modules is injective iff $R$ is Noetherian . Chase's theorem states that any direct product of projective $R$-modules is projective iff $R$ is Artinian . My question is : Is any characterization for commutative rings (with unity) are known such that any countable direct-product of projective modules over the ring is again projective ? 


Also asked on MSE : <https://math.stackexchange.com/questions/2501945/when-is-countable-direct-product-of-projective-modules-again-projective>



When is countable direct-product of projective modules again projective ?
Under the assumption that a *countable direct product of modules over $R$* means a direct product of countably many modules over $R$, I answer OP's question when $R$ is **Noetherian**. In addition, I outline a remark for $R$ an arbitrary **countable** ring with identity. 



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> > **Claim 1.** Let $R$ be a commutative ring with identity. If $R$ is Noetherian then the following are equivalent:
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> $(i)$ Every direct product of countably many projective modules over $R$ is projective.
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> $(ii)$ $R$ is Artinian.
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> > *Proof*. The implication $(ii) \Rightarrow (i)$ is given by Stephen Chase's Theorem [1, Theorems 3.3 and 3.4]. Assume that $(i)$ holds. Then $M \Doteq R^{\aleph\_0}$ is a projective module. If $R$ is moreover connected, then $M$ is free over $R$ by [2, Corollary 4.5]. $R$. By John O'Neill's result [3, Lemma 1.1], the ring $R$ is then Artinian. If $R$ is not connected, it is the direct product of finitely many non-trivial connected rings $R\_i$. By hypothesis, the module $M\_i \Doteq R\_i^{\aleph\_0}$ is projective over $R$, and hence over $R\_i$, for every $i$. As $R$ is a product of finitely many Artinian rings, it is Artinian. 
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For an arbitrary countable ring, the following weaker result holds.



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> > **Claim 2.** If $R$ is a countable ring with identity such that $R^{\aleph\_0}$ is projective over $R$, then $R$ is left [perfect](https://en.wikipedia.org/wiki/Perfect_ring).
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> > *Proof.* Apply [1, Theorem 3.1] and Bass's characterization of left perfect rings.
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Coming back to the commutative setting, we get



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> > **Corollary.**
> >  Let $R$ be an integral domain. If $R$ is countable then the following are equivalent:
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> $(i)$ Every direct product of countably many projective modules over $R$ is projective.
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> $(ii)$ $R$ is a field.
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> > *Proof.* Any element of a perfect commutative ring is either a unit or a zero divisor. The result is then a direct consequence of Claim 2.
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[1] S. Chase, ["Direct product of modules"](http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120260-3/S0002-9947-1960-0120260-3.pdf), 1960.  

[2] H. Bass, ["Big projective modules are free"](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.455.5945&rep=rep1&type=pdf), 1962.  

[3] J. O'Neill, ["When a ring is an $F$-ring"](https://ac.els-cdn.com/S0021869383710744/1-s2.0-S0021869383710744-main.pdf?_tid=2e77e806-c648-11e7-874f-00000aacb35e&acdnat=1510340078_14c303ff81fae85e7334cbe40c7f0106), 1993. 


