Completely Centrally Essential Rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913757912891392 |
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| author | Lyubimtsev, Oleg Tuganbaev, Askar |
| author_facet | Lyubimtsev, Oleg Tuganbaev, Askar |
| contents | A ring $R$ is said to be centrally essential if for every its non-zero element $a$, there exist non-zero central elements $x$ and $y$ with $ax = y$. A ring $R$ is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such a ring is a $PI$-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If $R$ is a commutative domain and $G$ is an arbitrary group, then any completely centrally essential group ring $RG$ is commutative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completely Centrally Essential Rings Lyubimtsev, Oleg Tuganbaev, Askar Rings and Algebras 16D25, 16R99 A ring $R$ is said to be centrally essential if for every its non-zero element $a$, there exist non-zero central elements $x$ and $y$ with $ax = y$. A ring $R$ is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such a ring is a $PI$-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If $R$ is a commutative domain and $G$ is an arbitrary group, then any completely centrally essential group ring $RG$ is commutative. |
| title | Completely Centrally Essential Rings |
| topic | Rings and Algebras 16D25, 16R99 |
| url | https://arxiv.org/abs/2503.20009 |