On the structure of elementary nets over quadratic field

The structure of elementary nets over quadratic fields is studied. A set of additive subgroups σ=(σij), 1≤i,j≤n, of a ring R is called a net of order n over R if σirσrj⊆σij for all i, r, j. The same system, but without the diagonal, is called elementary ne (elementary carpet). An elementary net σ=(σij) is called irreducible if all additive subgroups σij are different from zero. Let K=Q(√d) be a quadratic field, D a ring of integers of the quadratic field K, σ=(σij) an irreducible elementary net of order n≥3 over K, and σij a D-modules. If the integer d takes one of the following values (22 fields): -1, -2, -3, -7, -11, -19, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73, then for some intermediate subring P, D⊆P⊆K, the net σ is conjugated by a diagonal matrix of D(n,K) with an elementary net of ideals of the ring P.