Cyclically presented modules, projective covers and factorizations

TL;DR

This paper characterizes rings over which all cyclically presented modules have projective covers, linking this property to Von Neumann regularity modulo the Jacobson radical and idempotent lifting.

Contribution

It provides a characterization of rings where cyclically presented modules always have projective covers, connecting module theory with ring regularity and idempotent lifting properties.

Findings

Rings where every cyclically presented module has a projective cover are Von Neumann regular modulo J(R).

Idempotents can be lifted modulo J(R) in these rings.

Applications to modules with endomorphism rings that are regular modulo J(E).

Abstract

We investigate projective covers of cyclically presented modules, characterizing the rings over which every cyclically presented module has a projective cover as the rings $R$ that are Von Neumann regular modulo their Jacobson radical $J(R)$ and in which idempotents can be lifted modulo $J(R)$. Cyclically presented modules naturally appear in the study of factorizations of elements in non-necessarily commutative integral domains. One of the possible applications is to the modules $M_R$ whose endomorphism ring $E:=(M_R)$ is Von Neumann regular modulo $J(E)$ and in which idempotents lift modulo $J(E)$.