Subnormalizing extensions and D-structures

If S is a subnormalizing extension [7] of a ring R with identity generated by a set which is a multiplicative submonoid of S and generates S as a free unital left R-module, then using the left and right module structures of S one can define a set of mappings R → R which satisfy all but (possibly) one of the requirements for a D-structure [2], [3], [4]. It remains unknown whether this remaining condition must in fact be satisfied, but we show that it is satisfied for a number of particular monoids and is at least partially satisfied in general. In the other direction it is known that many but by no means all D-structures are linked to subnormalizing extensions in this way.