Upper triangular matrices of graded division algebras and their identities

In the present paper we study UT(D1, ... , Dn), a G-graded algebra of block triangular matrices where G is a group and the diagonal blocks D1, ... , Dn are graded division algebras. We prove that any two such algebras are G-isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of UTn to the case in which the diagonal blocks D1, ... , Dn are all isomorphic.