We study the electronic structure of a magnetic quantum ring formed by inhomogeneous magnetic fields, where electrons are confined to a plane, and the magnetic fields are zero inside the ring and constant elsewhere. The energy states that deviate from the Landau levels are found to form the magnetic edge states along the boundary regions of the magnetic quantum ring. The probability densities of these magnetic edge states are found to be well corresponded to the circulating classical trajectories. In contrast to magnetic or conventional quantum dots, the eigenstates of the magnetic quantum ring show angular momentum transitions in the ground state as the magnetic field increases, even without including electron-electron interactions. For a modified magnetic quantum ring with the distribution of nonzero magnetic fields inside the ring and different fields outside it, we also find similar behaviors such as the angular momentum transitions in the ground state with increasing the magnetic field.