The creation of a magnetic field in a ring may need to be accessed by means of an air gap. This is typically used to apply the magnetic field to a current carrying conductor to produce a force - e.g. in an electric motor.]
Consider therefore, a ring with an air gap present.
Note that since the flux lines are continuous, the flux density in the air gap and the ring will be the same.
However, the air gap represents a much larger reluctance to the flux than the high µ ring, so for the same magneto-motive force, the flux density in the ring will be reduced significantly compared to the case when no air gap was present.