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!
2026-03-26 00:58:55.1774486735
Matrices over noncommutative rings
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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.