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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2505.13647 |
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| _version_ | 1866908373267513344 |
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| author | Taherifar, A. |
| author_facet | Taherifar, A. |
| contents | 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 $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $z^\circ$-ideals and annihilator ideals Taherifar, A. General Topology 16D25, 16D70, 06D22 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 $R$. |
| title | On $z^\circ$-ideals and annihilator ideals |
| topic | General Topology 16D25, 16D70, 06D22 |
| url | https://arxiv.org/abs/2505.13647 |