Supercentralizing Maps in Prime Superalgebras

Abstract

Let A be a prime superalgebra over a commutative ring F with and f:A → A a supercentralizing F-linear map on A. We show that there exist an element λ in the extended centroid C of A and an F-linear map μ:A → C such that f(x) = λ x + μ(x) for all x ∈ A. This gives a version of Brešar's theorem for superalgebras. As a consequence, we show that a nontrivial prime superalgebra admitting a “nontrivial” supercentralizing F-linear map satisfies the standard identity of degree 4.

Key Words:

• Graded-prime
• Graded-simple
• Superalgebra
• Supercenter
• Supercentralizing
• Supercommutator
• Trivial superalgebra

2000 Mathematics Subject Classification:

• 16N60
• 16N25
• 16W25

ACKNOWLEDGMENT

The authors would like to express their thanks to the referee for his/her valuable suggestions of shortening some of the proofs.

Notes

Communicated by M. Bresar.