Isomorphisms of formal matrix incidence rings

In 2008 P.A. Krylov showed that formal matrix rings $K_{s}(R)$ and $K_{t}(R)$ are isomorphic if and only if elements $s$ and $t$ differ by invertible element, up to automorphism. The same result was proved and for many different cases. This paper concerns formal matrix rings (and algebras) with the same structure as incidence rings. We show that isomorphism problem for formal matrix incidence rings can be reduced to isomorphism problem of generalized incidence algebras. It appears that straight part of Krylov's theorem holds for this algebras whilst the opposite is not true. In particular, full classification of isomorphisms of generalized incidence algebras of order 4 over a field is constructed. Also isomorphism problem for a special case of formal matrix rings is considered: formal matrix rings with zero trace ideals.