TITLE: Matrices over noncommutative rings
QUESTION [5 upvotes]: I am wondering whether matrices over noncommutative rings have gone undergone a systematic study, particularly noncommutative group rings? I would appreciate sources, if any are available. Thanks!

REPLY [5 votes]: I feel like the other answer suggests (probably not intentionally) that matrix rings over noncommutative rings don't appear often because they are not well-understood, but I think rather the opposite is true: you don't see a lot explicitly said because they are well-understood.
One interested in the subject should probably learn the basics of Morita equivalence of categories of modules. Essentially the theorems characterizes when the $R$-Mod and $S$-mod are equivalent (in a sense), and it does so in such a way that one can easily describe what rings are Morita equivalent to $R$: to get to any other Morita equivalent ring from $R$, you are allowed to go from $S\mapsto M_n(S)$, or from $S\mapsto eSe$ where $e$ is an element of $S$ such that $e^2=e$ and $SeS=S$.  So with suitable $n$ and $e$, you can reach any equivalent ring as $eM_n(R)e$.
So what I'm saying is this: much is known about $M_n(R)$ since it is a special case of Morita equivalence. Many properties are preserved under Morita equivalence (and some are not, be careful!) That means that quite often you know when $R$ shares properties with $M_n(R)$.
I'm not completely clear on what the OP's intent was when saying "particularly noncommutative group rings." Since it doesn't make a lot of sense to interpret it as "matrix rings, in particular group rings, " I am guessing it means "matrix rings over group rings."  I think the study of "matrix rings over group rings" is probably not a thing, but it is certainly interesting to study passage of properties from $R$ to $R[G]$ and from $R$ to $M_n(R)$ independently, and then you can chain the two sets of results together to learn what you can.