  In this note, we show that an uncountable locally free group, and therefore
every locally free group, has a free subgroup whose cardinality is the same as
that of $G$. This result directly improve the main result in [T.
Nishinaka,"Group rings of countable non-abelian locally free groups are
primitive", Int. J. algebra and computation, 21(3)(2011), 409-431] and
establish the primitivity of group rings of locally free groups.
