Inner and Outer Twisted Derivations of Cyclic Group Rings
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909358737063936 |
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| author | Manju, Praveen Sharma, Rajendra Kumar |
| author_facet | Manju, Praveen Sharma, Rajendra Kumar |
| contents | In this article, we study twisted derivations of cyclic group rings. Let $R$ be a commutative ring with unity, $G$ be a finite cyclic group, and ($σ, τ$) be a pair of $R$-algebra endomorphisms of the group algebra $RG$, which are $R$-linear extensions of the group endomorphisms of $G$. In this article, we give two characterizations concerning $(σ, τ)$-derivations of the group ring $RG$. First, we develop a necessary and sufficient condition for a $(σ, τ)$-derivation of $RG$ to be inner. Second, we provide a necessary and sufficient condition for an $R$-linear map $D: RG \rightarrow RG$ with $D(1) = 0$ to be a $(σ, τ)$-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for $RG$: Under what conditions are all $(σ, τ)$-derivations of $RG$ inner? Or is the space of outer $(σ, τ)$-derivations trivial? More precisely, we give a sufficient condition under which all $(σ, τ)$-derivations of $RG$ are inner and a sufficient condition under which $RG$ has non-trivial outer $(σ, τ)$-derivations. Our result helps in generating several examples of non-trivial outer derivations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inner and Outer Twisted Derivations of Cyclic Group Rings Manju, Praveen Sharma, Rajendra Kumar Rings and Algebras Group Theory Representation Theory In this article, we study twisted derivations of cyclic group rings. Let $R$ be a commutative ring with unity, $G$ be a finite cyclic group, and ($σ, τ$) be a pair of $R$-algebra endomorphisms of the group algebra $RG$, which are $R$-linear extensions of the group endomorphisms of $G$. In this article, we give two characterizations concerning $(σ, τ)$-derivations of the group ring $RG$. First, we develop a necessary and sufficient condition for a $(σ, τ)$-derivation of $RG$ to be inner. Second, we provide a necessary and sufficient condition for an $R$-linear map $D: RG \rightarrow RG$ with $D(1) = 0$ to be a $(σ, τ)$-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for $RG$: Under what conditions are all $(σ, τ)$-derivations of $RG$ inner? Or is the space of outer $(σ, τ)$-derivations trivial? More precisely, we give a sufficient condition under which all $(σ, τ)$-derivations of $RG$ are inner and a sufficient condition under which $RG$ has non-trivial outer $(σ, τ)$-derivations. Our result helps in generating several examples of non-trivial outer derivations. |
| title | Inner and Outer Twisted Derivations of Cyclic Group Rings |
| topic | Rings and Algebras Group Theory Representation Theory |
| url | https://arxiv.org/abs/2410.16279 |