  We extend the original work of Ruderman, Kittel, Kasuya, and Yosida (RKKY) on
the interaction between two magnetic moments embedded in an electron gas to the
case where the electron gas is spin polarized. The broken symmetry of a host
material %(here we have a broken time reversal symmetry) introduces the
Dzyaloshinsky-Moriya (DM) vector and tensor interaction terms, in addition to
the standard RKKY term, so that the net interaction energy has the form: $
{\cal H} = J \vec S_1 \cdot \vec S_2 + \vec D \cdot \vec S_1 \times \vec S_2 +
\vec S_1 \cdot \stackrel{\leftrightarrow}{\Gamma} \cdot \vec S_2 $.
  We find that for the spin-polarized electron gas, a non-zero tensor
interaction $\stackrel{\leftrightarrow}{\Gamma}$ is present in addition to the
scalar RKKY interaction $J$, while $\vec D$ is zero due to the presence of
inversion symmetry. Explicit expressions for these are derived for the electron
gas both in 2D and 3D. The RKKY interaction exhibits a beating pattern, caused
by the presence of the two Fermi momenta $k_{F\uparrow}$ and $k_{F\downarrow}$,
while the $R^{-3}$ distance dependence of the original RKKY result for the 3D
electron gas is retained.
  This model serves as a simple example of the magnetic interaction in systems
with broken symmetry, which goes beyond the RKKY interaction.
