Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.     

Rings over Which Matrices Are Sums of Idempotent and $ q $ -Potent Matrices

Siberian Mathematical Journal - We study the rings over which each square matrix is the sum of an idempotent matrix and a  $ q $ -potent matrix. We also show that if  $ F $ is...     

Isomorphisms of Formal Matrix Rings with Zero Trace Ideals

We obtain explicit criteria for the isomorphism of formal matrix rings with zero trace ideals. In particular, we consider the case of formal upper-triangular matrix rings with semicentral reduced rings on the principal diagonal.