The Hall effect is studied as a correction to ideal magnetohydrodynamics (MHD) in the context of how it affects the linear stability of the cylindrical pinch in the simple case of a static, homogeneous equilibrium state. The effects of compressibility and electron pressure are included. The presence of the electron pressure gives rise to an electric field tangent to the boundary of the plasma. This introduces an additional boundary condition in the case of a perfectly conducting plasma boundary. Imposing this boundary condition eliminates the wave solutions presented in this paper. With respect to large radial wavenumber, the accumulation point of the slow magnetoacoustic wave frequency spectrum is changed from its finite value in ideal MHD to infinity by the Hall effect. The Hall effect gives rise to linear waves that do not exist in ideal MHD. Specifically, the Hall effect induces azimuthally symmetric, compressible, Alfve´n‐type wave propagation. The frequency spectrum of these waves is discrete and infinite and is a singular perturbation of the incompressible Alfve´n wave spectrum in ideal MHD. These Alfve´n‐type waves do not exist if the plasma is incompressible or ifH≡K⋅B=0, whereKis the wave vector of the perturbation andBis the equilibrium magnetic field.