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Cyclically presented modules, projective covers and factorizations

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 $J(E)$.