Kinetic stability properties of an intense relativistic electron ring in modified and conventional betatron configurations are investigated using the linearized Vlasov–Maxwell equations. Included is the important influence of intense equilibrium self‐fields. It is assumed that the ring is thin, and that &ngr;/&ggr;b≪1, where &ngr; is Budker’s parameter and &ggr;bmc2is the characteristic electron energy. The stability analysis is carried out for eigenfrequency &ohgr; close to harmonics of the cyclotron frequency &ohgr;czin the vertical betatron field. Also included in the analysis is the influence of transverse electromagnetic effects and surface‐wave perturbations. Dispersion relations for longitudinal perturbations are obtained, where it is assumed that the ring is located inside a perfectly conducting toroidal shell. There are several noteworthy points. First, transverse electromagnetic effects can completely stabilize the negative‐mass instability for sufficiently high‐current rings when betatronfocusing forces exceed defocusing self‐field forces (&ohgr;2cz>&ohgr;2pe/&ggr;2b). Second, for &ohgr;2cz>&ohgr;2pe/&ggr;2bwith no charge neutralization or stabilizing spread in canonical angular momentum ( f=0 and &Dgr;=0), surface‐wave instabilities can be completely stabilized at a sufficiently low transverse beam temperature. Third, for &ohgr;2cz<&ohgr;2pe/&ggr;2b, sufficiently low transverse temperature together with surface effects combine to drive a radial kink instability. Finally, the dispersion relation is analyzed numerically for parameters representative of the Naval Research Laboratory’s modified betatron (National Technical Information Service Document No. AD‐A108359/1) and the Los Alamos National Laboratory’s Liner Driven Ring Accelerator (National Technical Information Service Document No. DE84007994) and Phermex Injected Conventional Betatron (National Technical Information Service Document No. DE86002441/XAB). Detailed stability results are presented for the projected operating regimes of these devices, including the effects of canonical angular momentum spread, inverse aspect ratio, slowly varying accelerating fields, location of the conducting wall, and radial elongation of the minor ring cross section.