Let F be an infinite field of characteristic different from 2. We study the ∗-polynomial identities of the ∗-minimal algebra R = UT∗(F ⊕ F, F). We describe the generators of the *-polynomial identities of R and a linear basis of the relatively free algebra of R. When char.F = 0, these results allow us to provide a complete list of polynomials generating irreducible GL × GL-modules decomposing the proper part of the relatively free algebra of R. Finally, the ∗-codimension sequence of R is explicitly computed.

Let F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer.