Rudimentary structural matrix rings

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 [Formula: see text] full matrix ring over [Formula: see text], are provided.