Reveals equivalence between definitions of matrix rings in algebra, suggesting deeper insights into their structure.
The concept of structural matrix rings, studied by van Wyk starting in 1988, is one of the special types of subrings of a full matrix ring over a ring. Before this effort, Edwin Clark used matrix units to define subrings of a full matrix ring over a field. In this paper, we show that the two definitions are equivalent over a ring and we provide further characterizations of structural matrix rings. In addition, we characterize rudimentary structural matrix rings. The cardinals of, respectively, the set of all (non-isomorphic) structural matrix rings, and all (non-isomorphic) rudimentary structural matrix rings, which are subrings of the n × n full matrix ring over ℤ, are provided.
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Chun et al. (2026) studied this question.
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