A mass m is placed at point P which lies on the axis of a ring of mass M and radius R at a distance R from its centre. The gravitational force on mass m is 

The situation is as shown in the figure.

Gravitational force on an object of mass m at point P at distance d from centre O lying on the axis of the circular ring of radius R and mass M is given by

$\mathrm{F}=\frac{\mathrm{GMmd}}{{\left({\mathrm{R}}^2+{\mathrm{d}}^2\right)}^{3/2}}$

When d = R, then 

$\mathrm{F}=\frac{\mathrm{GMmR}}{{\left({\mathrm{R}}^2+{\mathrm{R}}^2\right)}^{3/2}}=\frac{\mathrm{GMmR}}{{\left(2{\mathrm{R}}^2\right)}^{3/2}}=\frac{\mathrm{GMmR}}{2^{3/2}{\mathrm{R}}^3}=\frac{\mathrm{GMm}}{2\sqrt{2}{\mathrm{R}}^2}$