On lattice of basic z-ideals

  For an f-ring  with bounded inversion property, we show that   , the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever  is a semiprimitive ring, , the set of all basic -ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring  with bounded inversion property, we prove that  is a complemented lattice and  is a semiprimitive ring if and only if  is a complemented lattice and  is a reduced ring if and only if the base elements for closed sets in the space  are open and  is semiprimitive if and only if the base elements for closed sets in the space  are open and  is reduced. As a result, whenever  (i.e., the ring of continuous functions), we have  is a complemented lattice if and only if  is a complemented lattice if and only if  is a -space.