We introduce a model of a passive optical cavity based on the two-dimensional Lugiato-Lefever
equation, with a localized pump carrying intrinsic vorticity S, and the cubic or cubic-quintic nonlinearity.
Up to S = 5, stable vortex-ring states (vortex pixels) are produced by a variational approximation
and in a numerical form. Surprisingly, vast stability areas of the vortex states are found, for
both the self-focusing and defocusing signs of the nonlinearity, in the plane of the pump-strength and
loss parameters. When the vortex-rings are unstable, they are destroyed by azimuthal perturbations
which break the axial symmetry. The results suggest new possibilities for mode manipulations in
nonlinear optical media.