Element absorb Topology on rings
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911979771265024 |
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| author | Shahidikia, Ali |
| author_facet | Shahidikia, Ali |
| contents | In this paper, we introduce a new Topology related to special elements in a noncummutative rings. Consider a ring $R$, we denote by $\textrm{Id}(R)$ the set of all idempotent elements in $R$. Let $a$ is an element of $R$. The element absorb Topology related to $a$ is defined as $τ_a:=\{ I\subseteq R | Ia \subseteq I\} \subseteq P(R)$. Since this topology is obtained from act of ring, it explains Some of algebraic properties of ring in Topological language .In a special case when $e$ ia an idempotent element, $τ_e:=\{ I\subseteq R | Ie \subseteq I\} \subseteq P(R)$. We present Topological description of the pierce decomposition $ R=Re\oplus R(1-e)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Element absorb Topology on rings Shahidikia, Ali Rings and Algebras In this paper, we introduce a new Topology related to special elements in a noncummutative rings. Consider a ring $R$, we denote by $\textrm{Id}(R)$ the set of all idempotent elements in $R$. Let $a$ is an element of $R$. The element absorb Topology related to $a$ is defined as $τ_a:=\{ I\subseteq R | Ia \subseteq I\} \subseteq P(R)$. Since this topology is obtained from act of ring, it explains Some of algebraic properties of ring in Topological language .In a special case when $e$ ia an idempotent element, $τ_e:=\{ I\subseteq R | Ie \subseteq I\} \subseteq P(R)$. We present Topological description of the pierce decomposition $ R=Re\oplus R(1-e)$. |
| title | Element absorb Topology on rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2408.03300 |