Superderivations with invertible values

Abstract

Communicated by I. P. ShestakovWe examine the structure of unital associative superalgebras A having nonzero superderivations with zero or invertible values. Under some mild assumptions, we show that such a superalgebra A is either a division superalgebra D, or M 2 (D), or it is a local superalgebra with a unique maximal graded ideal M such that M 2 = (0). We also describe in details the local superalgebras that are possible.

Derivations with invertible values in flexible algebras

“…In [12] Komatsu and Nakajima described associative rings that allow generalized derivations with invertible values. The case of associative superalgebras with derivations with invertible values was studied in the paper of Demir, Albas, Argac and Fosner [5]. Nonassociative algebras admitting derivations with invertible values are described in the paper of Kaygorodov, Lopatin and Popov [11], where it was proved that Jordan algebra can be represented as a symmetric bilinear form J(V, f ) and as a division algebra of Albert type.…”

Jordan algebras admitting derivations with invertible values

“…In the paper [13] Komatsu and Nakajima described associative rings that allow generalized derivations with invertible values. The case of associative superalgebras with derivations with invertible values was studied in the paper of Demir, Albas, Argac, and Fosner [4]. The description of non-associative algebras addmiting derivations with invertible values began in paper of Kaygorodov and Popov [11], where it was proved that every alternative (nonassociative) algebra addmiting derivation with invertible values is a Cayley-Dickson over their center or a factor-algebra of polynomial algebra C[x]/(x 2 ) over a Cayley-Dickson division algebra.…”