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In stackexchange the questioner says that when R is a commutative ring with unity, the Chase's theorem states that any direct product of projective R-modules is projective iff R is Artinian.

I doubt the above statement because from SC60,Theorem 3.3. we know that when R is a commutative ring with unity, the Chase's theorem states that any direct product of projective R-modules is projective if and only if R is perfect and coherent. Is a commutative perfect and coherent ring an commutative Artin ring?

In the Proposition 5.3 of the paper in Communications in Algebra or in arXiv say that if R is a commutative perfect ring, then R is Artinian if and only if it is (1,1)-coherent. Is a (1,1)-coherent ring a coherent ring?

Liang Chen
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Yes, it is true. It's even given as Exercise 2 p 161 of Lam's Lectures on modules and rings.

Roughly speaking this is because in the commutative case, the perfect ring splits into local perfect rings with nilpotent maximal ideals, and this allows one to conclude the powers of that maximal ideal form a composition series of finite length (proving the local ring is Artinian.)

By my reading, regular coherence is $(\aleph_0,\aleph_0)$ coherence, in the terms of that paper. In the proof of the above problem, one step is noting that $R$ has a minimal ideal, and that $0\to M\to R\to R/M\to 0$ is a finite presentation of that ideal, so $M$ is finitely generated. To do this, we would just need to know that the $1$-generated submodule (the minimal ideal) of the $1$ geneated module $R$ is f.p., so $(1,1)$ coherence suffices (but is not the same thing as full-blown coherence, always.)

rschwieb
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