Lie maps on alternative rings

On ultra algebras

“…In this paper we have introduced Jordan ultra algebras, and then we have the following question: What is the connection between the structure of ultra algebras and Jordan ultra algebras? In addition, the study about map structures on nonassociative algebras has become an area of great interest of pure math in the last years, we can quote some recent works [1,2,3,4,5,6,7]. Therefore other line of research that appears here is to know when a map is additive on nonassociative ultra algebras.…”

Lie (Jordan) centralizers on alternative algebras

“…Further this problem was studied by Ferreira and Ferreira [10,9] for Jordan (triple) derivable map on alternative rings. Later on many authors studied the different maps on alternative rings or algebras see [32,18,17,31,30] and references therein. Centralizers on rings as well as algebras have been extensively investigated by many mathematicians see [6,5,4,1,2,3] and references therein.…”

Nonlinear *-Jordan-type derivations on alternative *-algebras

“…The study on characterizations of certain maps on non-associative algebraic structures has become an active and broad line of research in recent years, we can mention [2,5,6,7,8,9,10,11]. For the case of associative structures Brešar and Fošner in [1,12], presented the following definition: for a, b ∈ R, where R is a * -ring, we denote by {a, b} * = ab + ba * and [a, b] * = ab − ba * the * -Jordan product and the * -Lie product, respectively.…”

“…11 ) + ϕ(b 12 ) + ϕ(c 21 ) ∈ A ′ there exists T ∈ A such that ϕ(t) = ϕ(a 11 ) + ϕ(b 12 ) + ϕ(c 21 ), with t = t 11 + t 12 + t 21 + t 22 . Now, observing that q n * (e 2 , ..., e 2 , a 11 ) = 0 and using Proposition 1.2 and Lemma 2.3, we obtain ϕ(q n * (e 2 , ..., e 2 , t)) = ϕ(q n * (e 2 , ..., e 2 , a 11 )) + ϕ(q n * (e 2 , ..., e 2 , b 12 ))By injectivity of ϕ we have q n * (e 2 , ..., e 2 , t) = q n * (e 2 , ..., e 2 , b 12 ) + q n * (e 2 , ..., e 2 , c 21 ), that is, 2t 22 + t 12 + t 21 = b 12 + c 21 .…”