Multiplicative Lie derivation of triangular 3-matrix rings

2020/01/02 by Chen, Zhenhui, Hou, Jinchuan
#15A78 #16W25 #47B47 #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2001.00427

Abstract

A map ϕ on an associative ring is called a multiplicative Lie derivation if ϕ([x,y])=[ϕ(x),y]+[x,ϕ(y)] holds for any elements x,y, where [x,y]=xy-yx is the Lie product. In the paper, we discuss the multiplicative Lie derivations on the triangular 3-matrix rings \mathcal T=\mathcal T₃(\mathcal R_i; \mathcal M_ij). Under the standard assumption Q_i\mathcal Z(\mathcal T)Q_i=\mathcal Z(Q_i\mathcal T Q_i), i=1,2,3, we show that every multiplicative Lie derivation φ:\mathcal T→\mathcal T has the standard form φ=δ+γ with δ a derivation and γ a center valued map vanishing each commutator.

Related

• Characterization of Lie multiplicative derivation on alternative rings
• Lie n-multiplicative mapping on Triangular n-Matrix Rings
• Lie triple maps on generalized matrix algebras
• Lie higher derivations of arbitrary triangular algebras
• Multiplicative Lie-type derivations on alternative rings