Journal of Applied Mathematics
Volume 2012 (2012), Article ID 925092, 18 pages
http://dx.doi.org/10.1155/2012/925092
Research Article

Zero Triple Product Determined Matrix Algebras

Hongmei Yao1,2 and Baodong Zheng1

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2College of Science, Harbin Engineering University, Harbin 150001, China

Received 9 August 2011; Accepted 20 December 2011

Academic Editor: Xianhua Tang

Copyright © 2012 Hongmei Yao and Baodong Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑥 𝑦 𝑧 = 0 , then there exists a 𝒞 -linear operator 𝑇 ∶ 𝐴 3 → 𝑋 such that { 𝑥 , 𝑦 , 𝑧 } = 𝑇 ( 𝑥 𝑦 𝑧 ) 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 𝑀 𝑛 ( 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 𝑀 𝑛 ( 𝐹 ) , 𝑛 ≥ 3 , where F is any field with ch 𝐹 ≠ 2 , is also zero Jordan triple product determined.