An identity in prime superalgebras

Abstract

In this paper we investigate the functional identity in a prime associative superalgebras. We prove the following result. Suppose that there exists a nonzero additive mapping f = f₀+f₁, on a prime associative superalgebra A with char(R)≠2, satisfying the relation [f(x); y²] = 0 for all x, y ∈ H(A). If

• A is prime algebra then [f(A);A] = 0 or [A;A] = 0.
• A₀ is prime algebra then [f(A);A] = 0 and [A;A₀] = 0 or A is trivial. More-over, if C₁ = 0 then f₀(A₁) = 0 and f₁(A₀) = 0.

Mathematics Subject Classification (2010): 17A70, 17B01, 17B40.

Keywords: Superalgebra, prime superalgebra, semiprime superalgebra, superderivation, supercommutator, extended centroid