Jordan triple derivation on alternative rings

Abstract

Let D be a mapping from an alternative ring R into itself satisfyingUnder some conditions on R, we show that D is additive.

Jordan Triple Derivation on Alternative Rings

“…In this paper we shall continue the line of research introduced in refs. [6,7] where its authors demonstrate the following results.…”

“…In non-associative ring theory we can mention recent works such as [2][3][4][5] where the authors generalized the results for a class of non-associative rings, namely alternative rings. The present paper we investigate the problem of when a Jordan triple multiplicative derivation must be an additive map for the class of alternative rings.…”

On generalized ∗-reverse derivable maps

“…For the case of additivity of maps defined on non-associative rings and having a nontrivial idempotent, some results have already been proved. In alternative rings we can mention the works in [4], [5], [6], [7], [8], [9], [10], [11]. In light of all the cited papers, the natural question could be whether the results obtained for multiplicative derivations can also be discussed for multiplicative reverse derivations.…”

$*$-Lie-type maps on $C^*$-algebras

“…Thenceforth, diverse works have been published considering different types of associative and non-associative algebras. Among them we can mention [9,11,12,8,13,10,14,3]. In order to further develop the study of additivity of maps, the researches incorporated two new product into the theory, presented by Brešar and Fošner in [2,15], where the definition is as follows: for X, Y ∈ R , where R is a * −ring, we denote by X • Y = XY + Y X * and [X, Y ] * = XY − Y X * the * -Jordan product and the * -Lie product, respectively.…”