Abstract Algebra – Example of a Finite Non-Commutative Ring Without 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

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