Noncommutative ring

Remarks

Example: the ring $M_2(\R)$ of $2 \times 2$ real matrices is noncommutative. With $E_{12}$ the matrix with $1$ in position $(1,2)$ and $0$ elsewhere, and $E_{21}$ defined likewise, one has $E_{12} E_{21} = E_{11} \ne E_{22} = E_{21} E_{12}$. Non-example: the ring $\Z$ is not noncommutative, since $ab = ba$ for all integers $a, b$. Many authors use "noncommutative ring" for a ring that is not assumed to be commutative. In that use, commutative rings are allowed as special cases.