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Let $k$ be a field. Consider an infinite direct product of rings $\prod k$. This is an example of Von-Neumann regular ring (name also absolutely flat), that is, every module is flat.

I think this ring is nice! I have the following question:

1. Is $\prod k$ self-injective?

2. What is the global dimension of $\prod k$ ? Finite or infinite? I have the feeling that it is infinite.

I don't know how to deal with this question. Any help will be appreciated.

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  • $\begingroup$ I only know for sure it is $>1$ (otherwise it would be hereditary+self-injective, which implies Artinian.) . You can prove it has infinite global dimension if you show that the projective dimensions of its cyclic modules are unbounded. $\endgroup$
    – rschwieb
    May 4, 2018 at 13:37
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    $\begingroup$ Actually here I see it claimed that $\prod_{i=1}^\infty \mathbb C$ has global dimension $2$! $\endgroup$
    – rschwieb
    May 4, 2018 at 13:39
  • $\begingroup$ @rschwieb how to show if R is hereditary and self-injective,then R is Artinian?thanks $\endgroup$
    – Jian
    May 4, 2018 at 15:10
  • $\begingroup$ Osofsky, Barbara. "Rings all of whose finitely generated modules are injective." Pacific Journal of Mathematics 14.2 (1964): 645-650. Corollary to the main Theorem. It seems to be open access at Project Euclid $\endgroup$
    – rschwieb
    May 4, 2018 at 15:15

2 Answers 2

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The answer to question $1$ is that any product of fields is self-injective.

Actually, any product of self-injective rings is self-injective. See Self-injective ring on the Encyclopedia of Mathematics.

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The global dimension of $\prod_\kappa F$ depends on the cardinalities $\kappa$ and $|F|$. It can be both finite or infinite. You will find very interesting material on that topic on this MO post.

(And yes as previously noted, any product of right self-injective rings is right self-injective.)

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