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?



Finitely presented modules admitting projective covers
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.)**](http://dx.doi.org/10.1007/BF01406080), Invent. Math. 13, 295-304 (1971). [ZBL0232.16020](https://zbmath.org/?q=an:0232.16020).


*Azumaya, Goro*, [**F-semi-perfect modules**](http://dx.doi.org/10.1016/0021-8693(91)90065-G), J. Algebra 136, No. 1, 73-85 (1991). [ZBL0717.16005](https://zbmath.org/?q=an:0717.16005).


