3 (spelling) changed Noncomutative to Noncommutative

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.

2 added 21 characters in body; deleted 8 characters in body; deleted 11 characters in body

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

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.