  Even though it is broadly accepted that single Be stars are rapidly rotating
stars surrounded by a flat rotating circumstellar disk, there is still a debate
about how fast these stars rotate and also about the mechanisms involved in the
angular-momentum and mass input in the disk. We study the properties of stars
that rotate near their critical-rotation rate and investigate the properties of
the disks formed by equatorial mass ejections. We used the most recent Geneva
stellar evolutionary tracks for rapidly rotating stars that reach the critical
limit and used a simple model for the disk structure. We obtain that for a 9
Msun star at solar metallicity, the minimum average velocity during the Main
Sequence phase to reach the critical velocity is around 330 km/s, whereas it
would be 390 km/s at the metallicity of the Small Magellanic Cloud (SMC). Red
giants or supergiants originating from very rapid rotators rotate six times
faster and show N/C ratios three times higher than those originating from
slowly rotating stars. This difference becomes stronger at lower metallicity.
It might therefore be very interesting to study the red giants in clusters that
show a large number of Be stars on the MS band. On the basis of our single-star
models, we show that the observed Be-star fraction with cluster age is
compatible with the existence of a temperature-dependent lower limit in the
velocity rate required for a star to become a Be star. The mass, extension, and
diffusion time of the disks produced when the star is losing mass at the
critical velocity, obtained from simple parametrized expressions, are not too
far from those estimated for disks around Be-type stars. At a given
metallicity, the mass and the extension of the disk increase with the initial
mass and with age on the MS phase. Denser disks are expected in low-metallicity
regions.
