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We analyze the ordered state of nuclear spins embedded in an interactingtwo-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI).Stability of the ferromagnetic nuclear-spin phase is governed by nonanalyticdependences of the electron spin susceptibility $\chi^{ij}$ on the momentum($\tilde{\mathbf{q}}$) and on the SOI coupling constant ($\alpha$). The uniform($\tq=0$) spin susceptibility is anisotropic (with the out-of-plane component,$\chi^{zz}$, being larger than the in-plane one, $\chi^{xx}$, by a termproportional to $U^2(2k_F)|\alpha|$, where $U(q)$ is the electron-electroninteraction). For $\tq \leq 2m^*|\alpha|$, corrections to the leading,$U^2(2k_F)|\alpha|$, term scale linearly with $\tq$ for $\chi^{xx}$ and areabsent for $\chi^{zz}$. This anisotropy has important consequences for theferromagnetic nuclear-spin phase: $(i)$ the ordered state--if achieved--is ofan Ising type and $(ii)$ the spin-wave dispersion is gapped at $\tq=0$. Tosecond order in $U(q)$, the dispersion a decreasing function of $\tq$, andanisotropy is not sufficient to stabilize long-range order. However,renormalization in the Cooper channel for $\tq\ll2m^*|\alpha|$ is capable ofreversing the sign of the $\tq$-dependence of $\chi^{xx}$ and thus stabilizingthe ordered state. We also show that a combination of the electron-electron andSO interactions leads to a new effect: long-wavelength Friedel oscillations inthe spin (but not charge) electron density induced by local magnetic moments.The period of these oscillations is given by the SO length $\pi/m^*|\alpha|$.

Ferromagnetic order of nuclear spins coupled to conduction electrons: a combined effect of the electron-electron and spin-orbit interactions