Zero Triple Product Determined Matrix Algebras

Hongmei Yao,Baodong Zheng

doi:10.1155/2012/925092

Abstract

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.

Highlights

Over the last couple of years, several papers characterizing bilinear maps on algebras through their action on elements whose certain product is zero have been written; see 1–6
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 zero Jordan triple product determined
In 1, in order to determine whether a linear map preserving zero product resp., zero Jordan product, zero Lie product is “closed” to a homomorphism, the authors introduced the definitions of zero product resp., zero Jordan product, zero Lie product determined algebras

Summary

Introduction

Over the last couple of years, several papers characterizing bilinear maps on algebras through their action on elements whose certain product is zero have been written; see 1–6. The core idea of these definitions is to answer the aforementioned questions by determining the bilinear maps preserving zero product resp., zero Jordan product, zero Lie product. It follows another interesting preserver problem: whether a linear map preserving zero triple product resp., zero Jordan triple product is still “closed” to a homomorphism? We call that A is a zero triple product determined algebra if for every C-module X and every C-trilinear map {·, ·, ·}, a implies b. Mn B of n × n matrices over a unital algebra B is still zero triple product resp., zero Jordan triple product determined. The purpose of this paper is to characterize the zero triple product resp., zero Jordan triple product determined algebra under some additional restrictions on the unital algebra B. We end this section by giving an equivalent condition of b in the previous definition which is more convenient to use:

Zero Triple Product Determined Matrix Algebras

Zero Jordan Triple Product Determined Matrix Algebras