We study the time evolution of an incompressible fluid (both inviscid and viscous) with axial symmetry without swirl, when the initial vorticity is very concentrated in N disjoint rings. We show that in a suitable limit, in which the thickness of the rings (and the viscosity, in the viscous case) tends to zero, the vorticity remains concentrated in N disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.