  A topological space is called self-covering if it is a nontrivial cover of
itself. We prove that a closed self-covering manifold $M$ with free abelian
fundamental group fibers over a circle under certain assumptions. In
particular, we give a complete answer to the question whether a self-covering
manifold with fundamental group $\mathbb Z$ is a fiber bundle over $S^1$,
except for the $4$-dimensional smooth case. As an algebraic Hilfssatz, we
develop a criterion for finite generation of modules over a commutative
Noetherian ring. We also construct examples of self-covering manifolds with
non-free abelian fundamental group, which are not fiber bundles over $S^1$
