# Finitely presented modules admitting projective covers

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?

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.

• 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." – rschwieb Nov 5 '20 at 14:05
• @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. – Jeremy Rickard Nov 5 '20 at 14:09
• cambridge.org/core/services/aop-cambridge-core/content/view/… seems a good reference – Benjamin Steinberg Nov 5 '20 at 17:12
• Semiregular rings show up quite a bit in the study of exchange rings. – Pace Nielsen Nov 5 '20 at 18:12