# Global dimension of $\prod k$

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.

• 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. May 4, 2018 at 13:37
• Actually here I see it claimed that $\prod_{i=1}^\infty \mathbb C$ has global dimension $2$! May 4, 2018 at 13:39
• @rschwieb how to show if R is hereditary and self-injective,then R is Artinian?thanks
– Jian
May 4, 2018 at 15:10
• 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 May 4, 2018 at 15:15

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