i'm trying to construct a non-commutative ring with $5^9$ elements. Usually in questions like this I do something like start off with $\mathbb{Z_5}[x]$ and then mod out by some degree $9$ (or degree $10?$) irreducible polynomial to get a field with $5^9$ elements. The problem is the question wants it to be non-commutative! and so i am quite lost. Input is appreciated!!
Further more, if anyone could give me some examples of Euclidea rings other $\mathbb{Z}$ and $\mathbb{Z[x]}$ i'd appreciate that also!
In general if $R$ is a ring with $m$ elements, $M_n(R)$ has $m^{n^2}$ elements.
So just use $M_3(\mathbb Z_5)$.
There are tons of examples at the wiki page for Euclidean domains. Here is the DaRT search, whose content may grow over time.
For future reference, it is better to ask questions that are completely different from each other in different posts.