Nuclear rings at centers of barred galaxies exhibit strong star formation activities .
They are thought to undergo gravitational instability when sufficiently massive .
We approximate them as rigidly-rotating isothermal objects and investigate their gravitational instability .
Using a self-consistent field method , we first construct their equilibrium sequences specified by two parameters : \alpha corresponding to the thermal energy relative to gravitational potential energy , and \widehat { R } _ { B } measuring the ellipticity or ring thickness .
Unlike in the incompressible case , not all values of \widehat { R } _ { B } yield an isothermal equilibrium , and the range of \widehat { R } _ { B } for such equilibria shrinks with decreasing \alpha .
The density distributions in the meridional plane are steeper for smaller \alpha , and well approximated by those of infinite cylinders for slender rings .
We also calculate the dispersion relations of nonaxisymmetric modes in rigidly-rotating slender rings with angular frequency \Omega _ { 0 } and central density \rho _ { c } .
Rings with smaller \alpha are found more unstable with a larger unstable range of the azimuthal mode number .
The instability is completely suppressed by rotation when \Omega _ { 0 } exceeds the critical value .
The critical angular frequency is found to be almost constant at \sim 0.7 ( G \rho _ { c } ) ^ { 1 / 2 } for \alpha \gtrsim 0.01 and increases rapidly for smaller \alpha .
We apply our results to a sample of observed star-forming rings and confirm that rings without a noticeable azimuthal age gradient of young star clusters are indeed gravitationally unstable .