Example of ring which is neither commutative nor unital

Question

Give an example of ring which is neither commutative nor unital.

I think, subring of matrix ring is neither commutative nor has a unit element.

Answer 1

I think your idea should work if you for example take all $2\times2$ matrices with entries from $2\mathbb Z$ (i.e. even integers).

Answer 2

The subring of the quaternions $\{a + bi + cj + dk\}$ containing the elements whose coefficients $a, b, c, d$ are all in $k \mathbb{Z}$ for any integer $k \geq 2$.

Any group ring $R[G]$ where the group $G$ is nonabelian and the ring $R$ has no unit.

Answer 3

Let $A$ be an infinite set and let $S$ be a ring with unit (not necessarily commutative). Let $R$ be the set of $A \times A$ matrices with entries in $S$ and with only a finite number of non zero entries. Then $R$, with the usual sum and multiplication of matrices, is a non commutative ring without unit.