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A transition-matrix theory for two-photon ionization processes in rare-gas atoms or isoelectronic ions is presented. Uncoupled ordinary differential equations are obtained for the radial functions needed to calculate the two-photon transition amplitude. The implications of these equations are discussed in detail. In particular, the role of correlations involving virtually excited electron pairs, which are known to be essential to the description of single-photon processes, is examined for multiphoton ionization processes. Additionally, electron scattering interactions between two electron-hole pairs are introduced into our transition amplitude in the boson approximation since these have been found important in two-photon ionization of xenon by L’Huillier and Wendin [J. Phys. B 20, L37 (1987)]. Application of our theory is made to two-photon ionization of the 3p subshell of argon below the one-photon ionization threshold. Our results are compared to previous calculations of McGuire [Phys. Rev. A 24, 835 (1981)], of Moccia, Rahman, and Rizzo [J. Phys. B 16, 2737 (1983)], and of Pindzola and Kelly [Phys. Rev. A 11, 1543 (1975)]. Results are presented for both circularly and linearly polarized photons. Among our findings are, firstly, that the electron scattering interactions, which have not been included in previous calculations for argon, produce a substantial reduction in the two-photon single-ionization cross section below the one-photon ionization threshold, which is in agreement with findings of L’Huillier and Wendin for xenon. Secondly, we find that de-excitation of virtually excited electron pairs by absorption of a photon is important for describing the interaction of the atom with the photon field, as in the case of single-photon ionization processes, but that further excitation of virtually excited electron pairs by the photon field has completely negligible effects, indicating a major simplification of the theory for higher-order absorption processes.