Global dimension of $\prod k$

Question

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.

Answer 1

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.)

Answer 2

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.