LET CF (X) BE THE SOCLE OF C (X) (I.E., THE SUM OF MINIMAL IDEALS OF C (X)). WE DEFINE LCF (X) = {FÎ C (X): `SF= X, WHERE SF IS THE UNION OF ALL OPEN SUBSETS U IN X SUCH THAT |U\Z (F) |<¥, LCF (X) IS CALLED THE LOCALLY SOCLE OF C (X) AND IT IS A Z-IDEAL OF C (X) CONTAINING CF (X). WE CHARACTERIZE SPACES X FOR WHICH THE EQUALITY IN THE RELATION CF (X) Í LCF (X) Í C (X) IS HOLD. WE DETERMINE THE CONDITIONS SUCH THATLCF (X) IS NOT PRIME IN ANY SUBRINGS OF C (X) WHICH CONTAINS THE IDEMPOTENTS OFX. WE INVESTIGATE THE PRIMNESS OF LCF (X) IN SOME SUBRINGS OF C (X).