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 (A(1), ..., A(m)) of finite dimensional G-simple algebras, we focus on the study of the factorability of the T-G-ideals Id(G)((UT (A(1), ..., A(m)),(alpha) over tilde)) of the G-graded upper block triangular matrix algebras UT (A(1), ..., A(m)) endowed with elementary G-gradings induced by some maps (alpha) over tilde. When G is a cyclic p-group we prove that the factorability of the ideal Id(G)((UT (A(1), ..., A(m)), (alpha) over tilde) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1, ..., A(m) as well to the existence of a unique isomorphism class of (alpha) over tilde -admissible elementary G-gradings for UT(A(1), ..., A(m)) Moreover, we present some necessary and sufficient conditions to the factorability of Ida ((UT(A(1), A(2)), (alpha) over tilde)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A(1) and A(2). (C) 2020 Elsevier Inc. All rights reserved.