Nonunital prime rings graded by ordered groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912678401802240 |
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| author | Lännström, Daniel Lundström, Patrik Öinert, Johan Wagner, Stefan |
| author_facet | Lännström, Daniel Lundström, Patrik Öinert, Johan Wagner, Stefan |
| contents | Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group $G$, if $S$ is what we call ideally symmetrically $G$-graded, then we show that there is a bijective correspondence between the $G$-graded prime ideals of $S$ and the $G$-prime ideals of $S_e$. We use this correspondence in the case where $G$ is ordered and $S$ is ideally symmetrically $G$-graded to show that $S$ is prime if and only if $S_e$ is $G$-prime. These results generalize classical theorems by Năstăsescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically $G$-graded subrings of group rings over fully idempotent rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26734 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonunital prime rings graded by ordered groups Lännström, Daniel Lundström, Patrik Öinert, Johan Wagner, Stefan Rings and Algebras 16D25, 16D80, 16W50, 16N60, 16S88, 16S35 Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group $G$, if $S$ is what we call ideally symmetrically $G$-graded, then we show that there is a bijective correspondence between the $G$-graded prime ideals of $S$ and the $G$-prime ideals of $S_e$. We use this correspondence in the case where $G$ is ordered and $S$ is ideally symmetrically $G$-graded to show that $S$ is prime if and only if $S_e$ is $G$-prime. These results generalize classical theorems by Năstăsescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically $G$-graded subrings of group rings over fully idempotent rings. |
| title | Nonunital prime rings graded by ordered groups |
| topic | Rings and Algebras 16D25, 16D80, 16W50, 16N60, 16S88, 16S35 |
| url | https://arxiv.org/abs/2510.26734 |