TITLE: Finitely presented modules admitting projective covers
QUESTION [6 upvotes]: A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely generated projective $R$-modules is Krull-Schmidt. I was wondering, and what about the rings $R$ such that every finitely presented $R$-module has a projective cover? Do these rings have a special name, and are there characterizations of these rings, just like there are for semi-perfect rings?

REPLY [4 votes]: Such rings were called "$F$-semiperfect", and more recently (thanks to rschweib for the information) "semiregular".
One characterization is that these are the rings $R$ such that $\bar{R}=R/J(R)$, the quotient by the Jacobson radical, is von Neumann regular and idempotents lift from $\bar{R}$ to $R$. This is analogous to the characterization of semiperfect rings as those for which $\bar{R}$ is semisimple and idempotents lift from $\bar{R}$ to $R$.
Some old references:
Oberst, Ulrich; Schneider, Hans-Jürgen, Die Struktur von projektiven Moduln. (The structure of projective modules.), Invent. Math. 13, 295-304 (1971). ZBL0232.16020.
Azumaya, Goro, F-semi-perfect modules, J. Algebra 136, No. 1, 73-85 (1991). ZBL0717.16005.