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.