We have already learned ideal of a ring. A lattice is an algebraic structure which looks like a ring for both have two operations. Similarly, a lattice also has a special structure named as the same as ideal. Definition 1. A subset I of a lattice L is an ideal if it is a sublattice of L and x ∈ I and a ∈ L imply that x ∩ a ∈ I. Specially, A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I...

The goal of this paper is study closed ideals and annihilators over the class distributive dual weakly complemented lattices(DDWCL's). algebraic structure ideals, dense are shown. connection between andannihilators in obtained.

We construct various examples of non-trivial closed ideals the compact-by-approximable algebra AX:=K(X)/A(X) on Banach spaces X failing approximation property. The include following: (i) if has cotype 2, Y type AX≠{0} and AY≠{0}, then AX⊕Y at least 2 ideals, (ii) there are subspaces X⊂ℓp for 4<p<∞ X⊂c0 such that AX contains a ideal, (iii) is space Z AZ an uncountable lattice having reverse orde...