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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2601.12599 |
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| _version_ | 1866910164792115200 |
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| author | Kinyon, Michael MacHale, Desmond |
| author_facet | Kinyon, Michael MacHale, Desmond |
| contents | Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on $x$. In this paper, in certain cases where $n$ is a fixed constant, we find equational proofs of each theorem. For the odd exponent cases $n = 2k+1$ of Jacobson's theorem, our main tool is a lemma stating that for each $x$, $x^k$ is central. For Herstein's theorem, we consider the cases $n=4$ and $n=8$, obtaining proofs with the assistance of the automated theorem prover Prover9. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Elementary proofs of ring commutativity theorems Kinyon, Michael MacHale, Desmond Rings and Algebras 16U80, 03B35 Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on $x$. In this paper, in certain cases where $n$ is a fixed constant, we find equational proofs of each theorem. For the odd exponent cases $n = 2k+1$ of Jacobson's theorem, our main tool is a lemma stating that for each $x$, $x^k$ is central. For Herstein's theorem, we consider the cases $n=4$ and $n=8$, obtaining proofs with the assistance of the automated theorem prover Prover9. |
| title | Elementary proofs of ring commutativity theorems |
| topic | Rings and Algebras 16U80, 03B35 |
| url | https://arxiv.org/abs/2601.12599 |