Central idempotents in group-graded rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917511346257920 |
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| author | Öinert, Johan |
| author_facet | Öinert, Johan |
| contents | Let $G$ be a group and let $R$ be a $G$-graded ring. We show that a nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if $G$ is torsion-free, then every central idempotent lies in the principal component of the grading. Our results generalize those of H. Bass and R. G. Burns from group rings to non-commutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20008 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Central idempotents in group-graded rings Öinert, Johan Rings and Algebras Group Theory 16W50, 16U40, 16U70, 16S34, 16S35, 16S36, 16S88 Let $G$ be a group and let $R$ be a $G$-graded ring. We show that a nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if $G$ is torsion-free, then every central idempotent lies in the principal component of the grading. Our results generalize those of H. Bass and R. G. Burns from group rings to non-commutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner rings. |
| title | Central idempotents in group-graded rings |
| topic | Rings and Algebras Group Theory 16W50, 16U40, 16U70, 16S34, 16S35, 16S36, 16S88 |
| url | https://arxiv.org/abs/2605.20008 |