  We consider a random walker on a ring, subjected to resetting at
Poisson-distributed times to the initial position (the walker takes the
shortest path along the ring to the initial position at resetting times). In
the case of a Brownian random walker the mean first-completion time of a turn
is expressed in closed form as a function of the resetting rate. The value is
shorter than in the ordinary process if the resetting rate is low enough.
Moreover, the mean first-completion time of a turn can be minimised in the
resetting rate. At large time the distribution of winding numbers does not
reach a steady state, which is in contrast with the non-compact case of a
Brownian particle under resetting on the real line. The mean total number of
turns (and the variance of the net number of turns) grow linearly with time,
with a proportionality constant equal to the inverse of the mean
first-completion time of a turn.
