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?
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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.
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3$\begingroup$ I think one will find through a google search that the more common name for what you are calling an "F-semiperfect ring" nowadays is "semiregular ring." $\endgroup$– rschwiebCommented Nov 5, 2020 at 14:05
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1$\begingroup$ @rschwieb Thanks, I didn't know that! The only reason I knew where to look at all was that I had some reason (that I've now forgotten) to look at Azumaya's paper many years ago. $\endgroup$ Commented Nov 5, 2020 at 14:09
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1$\begingroup$ cambridge.org/core/services/aop-cambridge-core/content/view/… seems a good reference $\endgroup$ Commented Nov 5, 2020 at 17:12
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1$\begingroup$ Semiregular rings show up quite a bit in the study of exchange rings. $\endgroup$ Commented Nov 5, 2020 at 18:12