On the factorability of the ideal of *-graded polynomial identities of minimal varieties of PI *-superalgebras

It has been recently proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed ⁎-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra, $A:=UT_{mathbb{Z}_2}^ast (A_1, ldots , A_m)$, equipped with a suitable elementary $mathbb{Z}_2$-grading and graded involution. Here we give necessary and sufficient conditions so that $Id_{mathbb{Z}_2}^ast (A)$ factorizes in the product of the ideals of ⁎-graded polynomial identities of its ⁎-graded simple components $A_i$.