Duo property for rings by the quasinilpotent perspective
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911886320074752 |
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| author | Harmanci, Abdullah Kurtulmaz, Yosum Ungor, Burcu |
| author_facet | Harmanci, Abdullah Kurtulmaz, Yosum Ungor, Burcu |
| contents | In this paper, we focus on the duo ring property via quasinilpotent elements which gives a new kind of generalizations of commutativity. We call this kind of ring qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others it is proved that if the Hurwitz series ring $H(R; α)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_03811 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Duo property for rings by the quasinilpotent perspective Harmanci, Abdullah Kurtulmaz, Yosum Ungor, Burcu Rings and Algebras In this paper, we focus on the duo ring property via quasinilpotent elements which gives a new kind of generalizations of commutativity. We call this kind of ring qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others it is proved that if the Hurwitz series ring $H(R; α)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1. |
| title | Duo property for rings by the quasinilpotent perspective |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2102.03811 |