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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!

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  • $\begingroup$ They are well understood. Lewis Rowen's two-value Ring Theory has stuff about them. The properties are pretty closely linked to the ring you start with. $\endgroup$
    – arsmath
    Sep 18 '18 at 20:11
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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.

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I only know of a usage of Rings of Matrices over skew fields for the classification of artinian simple and semisimple rings, namely the Artin-Wedderburn theorem.

The structure of non-commutative grouprings themselves is still researched, since the module theory of those encodes the theory of representations of the group on modules over the coefficient ring of the groupring, which is occupying several areas of mathematics since decades.

Since the structure of those grouprings are in general not fully understood, i doubt that there is a general theory of matrix rings over them, but I don't know.

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