This paper is the first of two in which the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi systems, the aim being to describe real metals and superconductivity. In this paper the authors concentrate on exact results and restrict ourselves to a ring approximation, applicable essentially to a high-density regime. They show that in the ground-state formalism the random phase approximation (RPA) can be formulated as a system of coupled, bilinear integral equations satisfied by functions associated with the so-called four-point functions of the system which provide a measure of the two-particle two-hole components in the true ground-state wavefunction. These equations are analysed in the dimensionless parameter formed by the ratio of the species masses and exact analytic solutions obtained. For Coulombic potentials (V₁₁V₂₂=V²₁₂) they show that the exact analytic solution is unique and obtain an expression for the correlation energy. For non-Coulombic potentials (V₁₁V₂₂ not=V²₁₂) they indicate how to obtain a possible analytic solution. An RPA-like treatment of the one- and two-body equations in the excited-state formalism is provided for completeness.