Ricerc@Sapienza

Let $F$ be a field of characteristic zero and $p$ a prime. In the present paper it is proved that a variety of $Z_p$-graded associative PI $F$-algebras of finite basic rank is minimal of fixed $Z_p$-exponent $d$ if, and only if, it is generated by an upper block triangular matrix algebra $UT_{Z_p}(A_1,ldots, A_m)$ equipped with a suitable elementary $Z_p$-grading, whose diagonal blocks are isomorphic to $Z_p$-graded simple algebras $A_1,ldots,A_m$ satisfying $dim_F (A_1 opluscdotsoplus A_m)=d$.

In the present paper it is 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 $ast$-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary $Z_2$-grading and graded involution.