I would like to find some examples of non-commutative, reduced, unity rings. The Wikipedia page https://en.wikipedia.org/wiki/Reduced_ring doesn't provide too many examples of reduced rings, let alone non-commutative ones. I tried to construct some myself, but I wasn't able to come up with anything since I am kind of new to ring theory.
Every product of division rings, at least one of which isn't commutative, is reduced. In, fact every noncommutative subring of such a ring is reduced also. These are far from polynomial rings over division rings because they are usually not domains.
Every polynomial ring over a division ring which isn't a field is also reduced.
Free algebras over a field are reduced, so the ring of non-commutative polynomials $K\langle x_1,\ldots,x_n\rangle$. Also, skew-polynomial rings are reduced, so the ring $K[x,\sigma]$ of polynomials in $x$ such that $x\lambda=\sigma(\lambda)x$, where $\sigma$ is a field automorphism of $K$.