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

TL;DR

This paper studies $z^{ ext{o}}$-ideals in non-commutative rings, characterizes them in various matrix and semiprime rings, and explores properties of the lattice of right annihilator ideals, revealing new structural insights.

Contribution

It introduces and characterizes $z^{ ext{o}}$-ideals in non-commutative rings and analyzes the lattice of right annihilator ideals, extending existing theory.

Findings

Characterizations of $z^{ ext{o}}$-ideals in matrix rings and semiprime rings.

The lattice of right annihilator ideals forms a frame in semiprime rings.

Identification of smallest and largest right annihilator ideals in $SA$-rings.

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 $R$.