This work is concerned with the transport properties of superconducting nanorings at extremely low temperatures in magnetic field. The goal of this work was to experimentally observe a prediction on the crossover from h/2e to h/e period oscillations of transition temperature when the size of the ring becomes small relative to the superconducting coherence length. Impurities in the aluminum nanostructures studied here hinder the direct observation of this crossover. However, the proper direction to take in future experiments on this subject has been evaluated and firmly established. Along the way, an interesting effect has been observed in the form of a high resistance state in superconducting nanorings. This is remarkable because it has a resistance higher than the normal state resistance even though it is superconducting. Therefore, it seems to be phase coherent even though it is resistive. There have been other similar observations in the literature in wires and disks but not rings. It can be explained in terms of nonequilibrium relaxation of quasiparticles near normal-superconductor interfaces that occur naturally in such constricted structures. The relevant physics of this is discussed.