Wikipedia has separate articles on perfect fields and perfect rings. The former has its own definition of perfect rings (of prime characteristic). The latter has a hatnote that says:
This article is about perfect rings as introduced by Hyman Bass. For perfect rings of characteristic $p$ generalizing perfect fields, see perfect field.
That seems to indicate that the two definitions of perfect rings are not related, and this is also my impression from the content. If so, I feel that the articles (especially the one on perfect fields) don’t make it clear enough that the same term is being used for two unrelated concepts. I want to edit them accordingly, but first I want to make sure that I’m not missing something.
The definition of a left or right perfect ring in the article on perfect rings is:
A left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy.
The definition of perfect rings of prime characteristic in the article on perfect fields is:
A ring of characteristic $p$ is called perfect if the Frobenius endomorphism is an automorphism.
My question is:
- Are these two definitions of a perfect ring related?
(A question that might be related, though I don’t think so: Perfect ring Vs perfect module)
An infinite product of copies of the field of two elements is perfect in the former sense but not Bass' sense.
$F_p[x]/(x^p)$ is perfect in Bass' sense but not the former sense since $r^p=0$ for all nonunits.
So I don't see any connection, and I wasn't aware of any to begin with.