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3 | (spelling) changed Noncomutative to Noncommutative | ||
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The following pearl by Jacobson can under no circumstances be left out from the list: Let $\mathbf{R}$ be a ring with center $\mathrm{Z}$. Let us suppose that you can find $n \in \mathbb{N}_{>1}$ such that $x^{n}-x \in \mathrm{Z}$ for every $x \in \mathbf{R}$. Then $\mathbf{R}$ is a commutative ring. A good place to learn more about results of this kind is Herstein's Noncomutative Noncommutative rings. |
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2 | added 21 characters in body; deleted 8 characters in body; deleted 11 characters in body | ||
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The following pearl by Jacobson can under no circumstances be left out from the list: Let $\mathbf{R}$ be a ring with center $\mathrm{Z}$ \mathrm{Z}$. Let us suppose that you can find $n \in \mathbb{N}_{>1}$ such that $x^{n}-x \in \mathrm{Z}$ for every $x \in \mathbf{R}$ and a fixed natural number $n$ greater than $1$. mathbf{R}$. Then $\mathbf{R}$ is a commutative ring. In Herstein's Noncomutative rings you can find a delightful treatment of A good place to learn more about results of this typekind is Herstein's Noncomutative rings. |
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1 | [made Community Wiki] | ||
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The following pearl by Jacobson can under no circumstances be left out from the list: Let $\mathbf{R}$ be a ring with center $\mathrm{Z}$ such that $x^{n}-x \in \mathrm{Z}$ for every $x \in \mathbf{R}$ and a fixed natural number $n$ greater than $1$. Then $\mathbf{R}$ is a commutative ring. In Herstein's Noncomutative rings you can find a delightful treatment of results of this type. |
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