  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 $J(E)$.
