On $z^{\circ }$-ideals and annihilator ideals

For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ }$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class of $z^{\circ }$-ideals in non-commutative rings. We provide characterizations of $z^{\circ }$-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 a 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 contained in an ideal $I$ of an $SA$-ring $R$.