For a ∈ R, let Pa denote the intersection of all minimal prime ideals of R containing a. An ideal I of a ring R is called a z◦-ideal if Pa ⊆ I for all a ∈ I. In this paper, we first investigate the class of z◦-ideals in non-commutative rings. We provide characterizations of z◦-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, and semiprime rings. Next, we explore new properties of the lattice rAnn(id(R)), the set of right annihilator ideals of R. We prove that rAnn(id(R)) forms an algebraic frame when R is semiprime and a coherent frame when R is a reduced ring. Furthermore, we characterize the smallest (resp., largest) right annihilator ideal containing (resp., contained in) an ideal I of an SA-ring R.
