  We study the effect of the electron-electron interaction on the weak
localization correction of a ring pierced by a magnetic flux. We compute
exactly the path integral giving the magnetoconductivity for an isolated ring.
The results are interpreted in a time representation. This allows to
characterize the nature of the phase coherence relaxation in the ring. The
nature of the relaxation depends on the time regime (diffusive or ergodic) but
also on the harmonics $n$ of the magnetoconductivity. Whereas phase coherence
relaxation is non exponential for the harmonic $n=0$, it is always exponential
for harmonics $n\neq0$. Then we consider the case of a ring connected to
reservoirs and discuss the effect of connecting wires. We recover the behaviour
of the harmonics predicted recently by Ludwig & Mirlin for a large perimeter
(compared to the Nyquist length). We also predict a new behaviour when the
Nyquist length exceeds the perimeter.
