A new set of boundary conditions for a resistive plasma that takes into account the effect of a nonideal rigid wall of finite conductivity and thickness, is derived. These conditions, in cylindrical geometry, take the form of a system of integrodifferential equations involving the perturbed plasma variables v1, B1, T1, at the inner surface of the rigid wall. This system of integrodifferential equations is numerically solved, using a numerically stable scheme, and is implemented in an existing linear one-dimensional resistive initial value code, the Ripple IVA, developed by Dibiase and Killeen [Comput. Phys. 24, 158 (1977)]. An approximate solution of the problem, valid in a neighborhood of the time origin, is also derived. Attention is then focused on the linear behavior of resistive tearing and interchange modes as a function of the ratio of plasma and wall conductivity as well as the magnetic Lundquist number S, especially for high-beta equilibria of the reversed field pinch (RFP) type.