TITLE: Example of a finite non-commutative ring without a unity
QUESTION [20 upvotes]: 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.

REPLY [4 votes]: 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.