From the Vlasov-fluid model, a set of approximate stability equations describing the stability of a cylindrically symmetric z–pinch is derived. The equations are derived in the limit of small gyroradius and include kinetic effects such as finite Larmor radius, particle drifts and resonant particles. In the limit of zero Larmor radius and short wavelengths, we apply the equations to the internal m = 0 and m = 1 modes. If the drift term mωD + kVD is neglected in the resonant denominators, we find stability criteria that are more optimistic than the corresponding stability criteria for perpendicular MHD. The neglect of the drift term is, however, not justified for the m = 1 mode, where mωD needs to be retained in order to preserve the property that this approximate model should have the same point of marginal stability as the exact Vlasov-fluid model. Retaining the drift terms, growth rates have been calculated for the m = 1 mode, for a constant-current-density equilibrium and for the Bennett equilibrium. For the Bennett profile, we obtain, when compared with perpendicular MHD, a substantial reduction in growth rate γ/γMHD ≈ 0·2.