On $z^\circ$-ideals and annihilator ideals
A. Taherifar
TL;DR
The paper extends $z^{\circ}$-ideals to arbitrary rings and develops a non-commutative theory through characterizations in extension rings, notably 2-by-2 generalized triangular rings and matrix rings, while connecting these ideals to right annihilator structures. It establishes that in semiprime rings $z^{\circ}$-ideals coincide with right $d$-ideals, and it provides concrete extensions and counterexamples in extension rings. A central contribution is showing that the lattice of right annihilator ideals, $rAnn(id(R))$, forms a frame for semiprime $R$ and a coherent frame for reduced $R$, with explicit description of extremal annihilator closures and behavior under sums and intersections, yielding structural insights for SA-rings and related classes. Collectively, these results advance the understanding of the interactions between prime-like intersections, annihilator theory, and lattice-theoretic properties in non-commutative ring settings.
Abstract
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 $R$.