Superalgebras with graded involution and star-graded colength bounded by 3

Nilpotent superderivations in prime superalgebras

“…Therefore, since R is free of m−1 r−1 -torsion, a 2r−1 = 0 so a is nilpotent of index 2r. So n ≡ 8 6 and with s = n 2 , a s+1 = 0, a s = 0 and by Proposition 3.1 a is ad-nilpotent of R of index n, case (5).…”

“…Superinvolutions in associative superalgebras have been a topic of great interest. We highlight the works of J. Laliena [21] on the description of the derived superalgebra [K, K] of a semiprime superalgebra with superinvolution, the papers [22] and [23] of J. Laliena and R. Rizzo on the extension of results of C. Lanski and S. Montgomery to associative superalgebras with superinvolution, and the recent works of T. S. do Nascimento, A. C. Vieira, A. Giambruno, A. Ioppolo, D. La Mattina, F. Martino ( [5], [10], [11], [12], [18]) on superinvolutions in superalgebras related to polynomial identities and related to the growth of certain substructures of the superalgebras.…”

“…Note that any element of K 1 has the form 0 B C 0 where B is a symmetric matrix and C is a skew-symmetric matrix in M r (C) with respect to the usual transposition.Let us consider a symmetric matrix B ∈ M r (C) with B r = 0 and B r−1 = 0 (it is shown in[20, Corollary 5] that for every r there exist symmetric nilpotent matrices in M r (C) of rank r − 1). Let 0 = λ ∈ R and let i denote the square root of -B + (λi) id i.e., (a 2 − λi) is nilpotent of index r.When r is odd a is an example for Theorem 4.4 (3), and when r is even a is an example for Theorem 4.4(5). Both cases are examples of elements of the form (2.b) of Theorem 3.2.…”