We investigate the connection between two recent investigations on flux-periodic effects in one-dimensional normal-metal rings with inelastic diffusion length larger than the size of the ring. Büttiker, Imry, and Landauer have pointed out that closed rings, driven by an external flux, act like superconducting rings with a Josephson junction, except that 2e is replaced by e. Gefen, Imry, and Azbel considered such a ring connected to current leads and found a flux-periodic electric resistance. We establish a connection between these Aharonov-Bohm-like effects by demonstrating that the transmission probability of the ring, which determines the electric resistance, exhibits resonances near the energies of the electronic states of the closed ring. It is the flux dependence of the resonances which gives rise to the strong oscillatory behavior of the electric resistance. © 1984 The American Physical Society.