Zero Triple Product Determined Matrix Algebras

Let A be an algebra over a commutative unital ring C. We say that A is zero triple product determined if for every C-module X and every trilinear map {⋅,⋅,⋅}, the following holds: if {x,y,z}=0 whenever xyz=0, then there exists a C-linear operator T:A3⟶X such that x,y,z=T(xyz) for all x,y,z∈A. If the ordinary triple product in the aforementioned definition is replaced by Jordan triple product, then A is called zero Jordan triple product determined. This paper mainly shows that matrix algebra Mn(B), n≥3, where B is any commutative unital algebra even different from the above mentioned commutative unital algebra C, is always zero triple product determined, and Mn(F), n≥3, where F is any field with chF≠2, is also zero Jordan triple product determined

Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products

Suppose m, n≥2 are positive integers. Let n be the space of all n×n complex upper triangular matrices, and let ϕ be an injective linear map on m⊗n. Then ϕ(A⊗B) is an idempotent matrix in m⊗n whenever A⊗B is an idempotent matrix in m⊗n if and only if there exists an invertible matrix P∈m⊗n such that ϕ(A⊗B)=P(ξ1(A)⊗ξ2(B))P-1,   ∀A∈m,   B∈n, or when m=n, ϕ(A⊗B)=P(ξ1(B)⊗ξ2(A))P-1,   ∀A∈m,   B∈m, where ξ1([aij])=[aij] or ξ1([aij])=[am-i+1, m-j+1] and ξ2([bij])=[bij] or ξ2([bij])=[bn-i+1, n-j+1]