Example of a finite non-commutative ring without a unity

Question

Give an example of a finite, non-commutative ring, which does not have a unity.

I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any help is appreciated.

Answer 1

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

Answer 2

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

Answer 3

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.

Answer 4

In the spirit of the answer by massy255: take the rng of strictly upper triangular $n\times n$ matrices over a finite field for $n\geq3$. This rng does not even have a nonzero subrng with a unit.