Commuting maps on alternative rings
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Commuting maps on alternative rings Bruno Leonardo Macedo Ferreira1
· Ivan Kaygorodov2
Received: 18 September 2020 / Accepted: 12 November 2020 © Università degli Studi di Napoli "Federico II" 2020
Abstract Suppose R is a 2,3-torsion free unital alternative ring having an idempotent element e1 (e2 = 1 − e1 ) which satisfies xR · ei = {0} ⇒ x = 0 (i = 1, 2). In this paper, we aim to characterize the commuting maps. Let ϕ be a commuting map of R so it is shown that ϕ(x) = zx + (x) for all x ∈ R , where z ∈ Z(R ) and is an additive map from R into Z(R ). As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative C ∗ -algebra with no central summands of type I1 . Keywords Commuting maps · Alternative rings · Peirce decomposition Mathematics Subject Classification 17D05 · 47B47
1 Introduction Let R be a unital ring not necessarily associative or commutative and consider the following convention for its multiplication operation: x y·z = (x y)z and x · yz = x(yz) for x, y, z ∈ R, to reduce the number of parentheses. We denote the associator of R by (x, y, z) = x y · z − x · yz for x, y, z ∈ R. And [x, y] = x y − yx is the usual Lie product of x and y, with x, y ∈ R.
This work was supported by FAPESP 19/03655-4; CNPq 302980/2019-9; RFBR 20-01-00030.
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Bruno Leonardo Macedo Ferreira [email protected] Ivan Kaygorodov [email protected]
1
Federal University of Technology, Professora Laura Pacheco Bastos Avenue, 800, Guarapuava 85053-510, Brazil
2
Federal University of ABC, dos Estados Avenue, 5001, Santo André 09210-580, Brazil
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B. L. M. Ferreira, I. Kaygorodov
Let be ϕ : R → R a additive map of R into R. We call ϕ a commuting map of R into R if for all x ∈ R: [ϕ(x), x] = 0. According to [22], Divinsky at [8] started a study on commuting maps, he proved that a simple Artinian ring is commutative if it has a commuting automorphism different from the identity map. In the face of this study, some mathematicians began to investigate the problem of characterizing some maps on associative rings. Bresar in [3] proved that, if F is a commuting additive map from a von Neumann algebra M into itself, then there exist Z ∈ Z(M) and an additive map h : M → Z(M) such that F(A) = Z A + h(A) for all A ∈ M. Later, Bresar [5] gave the same characterization of commuting additive maps on prime rings. For the reader interested in other results about characterization of maps on associative rings and algebras, see refs. [4,6,7,20,21] and the references therein. For the class of alternative rings and algebras studies about many linear and nonlinear maps have been an interesting and active research topic recently, we can quote [10–15,19]. An important class of 8-dimensional Cayley algebras (or Cayley–Dickson algebras, the prototype having been discovered in 1845 by Cayley and later generalized by Dickson) is so called octonions algebra a class of alternative algebras which are not associat
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