On the structure of nets over quadratic fields

The structure of nets over quadratic fields is studied. Let K=Q(d--√) be a quadratic field, D the ring of integers of the quadratic field K. A set of additive subgroups σ=(σij), 1≤i,j≤n, of a~field K is called a net of order n over K if σirσrj⊆σij for all values of the index i, r, j. A net σ=(σij) is called irreducible if all additive subgroups σij are different from zero. A net σ=(σij) is called a D-net if 1∈τii, 1≤i≤n. Let σ=(σij) be an irreducible D-net of order n ≥ 2 over K, where σij are D-modules. We prove that, up to conjugation diagonal matrix, all σij are fractional ideals of a fixed intermediate subring P, D⊆P⊆K, and all diagonal rings coincide with P: σ11=σ22=…=σnn=P, where σij⊆P are integer ideals of the ring P for any i j, then P⊆σij. For any i, j we have σ1j⊆σij.