On exponent matrices of tiled orders

A ring Λ is called a tiled order if Λ is a prime Noetherian semi-prefect semi-distributive ring with non-zero Jacobson radical. Any tiled order can be constructed from a (non-necessarily commutative) discrete valuation ring and an exponent matrix. The latter means a square integer matrix A = (αps), whose diagonal entries are equal to zero, and for all possible indices i, j, k, the following inequality holds: αij + αjk ≥ αik. Every tiled order Λ is isomorphic to a ring of the form