  Nuclear magnetism in n-doped semiconductors with positive hyperfine constant
is revisited. Two kinds of nuclear magnetic ordering can be induced by resident
electrons in a deeply cooled nuclear spin system. At positive nuclear spin
temperature below a critical value, randomly oriented nuclear spin polarons
similar to that predicted by I. Merkulov [I. Merkulov, Physics of the Solid
State 40, 930 (1998)] should emerge. These polarons are oriented randomly and
within each polaron nuclear and electron spins are aligned
antiferromagnetically. At negative nuclear spin temperature below a critical
value we predict another type of magnetic ordering - dynamically induced
nuclear ferromagnet. This is a long-range ferromagnetically ordered state
involving both electrons and nuclei. It can form if electron spin relaxation is
dominated by the hyperfine coupling, rather than by the spin-orbit interaction.
Application of the theory to the n-doped GaAs suggests that the ferromagnetic
order may be reached at experimentally achievable nuclear spin temperature
$\Theta_N \approx 0.5$ $\mu$K and lattice temperature $T_L \approx 5$ K.
