Zero Insertive Nil Clean Rings
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913972279574528 |
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| author | Subba, Sanjiv Subedi, Tikaram |
| author_facet | Subba, Sanjiv Subedi, Tikaram |
| contents | This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ and $ZI(R)$ denotes the set of all zero insertive elements of $R$. We establish that a ring $R$ is semicommutative if and only if $ZI(R) \subseteq E(R)$, and weakly semicommutative if and only if $ZI(R) \subseteq N(R)$, where $E(R)$ and $N(R)$ denote respectively the sets of idempotent elements and nilpotent elements. For ZINC rings with no nontrivial idempotents, $ZI(R) \subseteq N(R)$.
We prove that a finite direct product of ZINC rings is ZINC if and only if each component ring is ZINC, while an infinite direct product may fail to be ZINC. For $n \geq 2$, if $M_n(R)$ is ZINC, then $R$ is weakly clean, however, the converse is not true (e.g., $\mathbb{Z}$). Additionally, $M_n(K)$ is ZINC for a division ring $K$ if and only if $K \cong \mathbb{F}_2$. We, also, present a ZINC ring whose polynomial and power series extensions are not ZINC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero Insertive Nil Clean Rings Subba, Sanjiv Subedi, Tikaram Rings and Algebras 16S50, 16S36, 16U80, 16U99 This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ and $ZI(R)$ denotes the set of all zero insertive elements of $R$. We establish that a ring $R$ is semicommutative if and only if $ZI(R) \subseteq E(R)$, and weakly semicommutative if and only if $ZI(R) \subseteq N(R)$, where $E(R)$ and $N(R)$ denote respectively the sets of idempotent elements and nilpotent elements. For ZINC rings with no nontrivial idempotents, $ZI(R) \subseteq N(R)$. We prove that a finite direct product of ZINC rings is ZINC if and only if each component ring is ZINC, while an infinite direct product may fail to be ZINC. For $n \geq 2$, if $M_n(R)$ is ZINC, then $R$ is weakly clean, however, the converse is not true (e.g., $\mathbb{Z}$). Additionally, $M_n(K)$ is ZINC for a division ring $K$ if and only if $K \cong \mathbb{F}_2$. We, also, present a ZINC ring whose polynomial and power series extensions are not ZINC. |
| title | Zero Insertive Nil Clean Rings |
| topic | Rings and Algebras 16S50, 16S36, 16U80, 16U99 |
| url | https://arxiv.org/abs/2508.01333 |