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We study a superconducting Kitaev ring pierced by a magnetic flux, with andwithout disorder, in a quantum ring configuration, and in a rf-SQUID one, wherea weak link is present. In all cases we find in the ground state a finitesupercurrent in the thermodynamic limit. This current is periodic in the flux,with a period strictly associated with the chosen configuration. In therf-SQUID the supercurrent shows jumps at a fixed value $\Phi^*$ of the flux,when the system is topological. In the thermodynamic limit $\Phi^*$ is constantinside the topological phase, independently of disorder, and we analyticallypredict it using a perturbative approach in the weak-link coupling. We findthat the weak link breaks the ground-state degeneracy due to the boundaryMajorana modes, and opens a topological spectral gap for $\Phi\neq \Phi^*$,that vanishes at $\Phi^*$ with a cusp providing the current jump. Looking atthe quasiparticle excitations, we see that they are Anderson localized, so theycannot carry a resistive contribution to the current, and the localizationlength stays finite also for very small disorder strength, for a specific valueof the superconducting coupling. In the absence of disorder, we analyticallyand numerically find that the chemical-potential derivative of the supercurrentlogarithmically diverges at the topological/trivial transition, in agreementwith the transition being of the second order.

Topological properties of gap and supercurrent, and Anderson localization of quasiparticles, in a Kitaev ring threaded by a magnetic flux