Example of a finite non-commutative ring without a unity

There are many examples in this spirit: the $n\times n$ matrices over a finite field with bottom row zero.

$\textbf{Hint:}$ Matrix rings are a good example of non-commutative rings.

The easiest example of such a ring is to let $$ S=\{2 n\;|\; n \in \mathbb{Z}\} $$ and then consider the ring $M_n(S)$, the ring of $n \times n$ matrices with elements in $S$ (notice this does not include the identity matrix as $1 \notin S$). To get the finite example, instead, simply take $2\mathbb{Z}/2n\mathbb{Z}$ instead of the set $S$.

In fact, for every prime $p$, there is a noncommutative ring without unity of order $p^2$. Moreover, if a ring of such order had a unit it would also necessarily be commutative.