Noncommutative ring with eight elements and with unity I'm a bit confused about an exercise I read. Namely, T. Y. Lam's A First Course in Noncommutative Rings has the following on page $23$ Ex. $1.10$.
Let $p$ be a fixed prime. Show that there exists a noncommutative ring (with identity) of order $p^3$.
Well, this implies that there is a noncommutative ring with identity of order $8<16$, which contradicts with benh's answer given in Smallest non-commutative ring with unity . So have I understood correctly that there is a mistake in Lam's exercise?

Indeed, there is a noncommutative ring with unity of order $p^3$ for all prime $p$. Though I have not read benh's proof, it must have an error of some form. I will not bother to repeat the argument. See this paper for many various constructions related to your question as well as the proof that there is a noncommutative ring with unity of order $p^3$ for all prime $p$. 
