On the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups

Let F be an algebraically closed field of characteristic zero and G be an arbitrary finite cyclic group. In this paper, given an m-tuple (A1,…,Am) of finite dimensional G-simple algebras, we focus on the study of the factorability of the TG-ideals IdG((UT(A1,…,Am),α˜)) of the G-graded upper block triangular matrix algebras UT(A1,…,Am) endowed with elementary G-gradings induced by some maps α˜. When G is a cyclic p-group we prove that the factorability of the ideal IdG((UT(A1,…,Am),α˜) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1,…,Am, as well to the existence of a unique isomorphism class of α˜-admissible elementary G-gradings for UT(A1,…,Am). Moreover, we present some necessary and sufficient conditions to the factorability of IdG((UT(A1,A2),α˜)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A1 and A2.