Zero Triple Product Determined Matrix Algebras. (25th February 2012)

• Record Type: Journal Article
• Authors: Yao, Hongmei; Zheng, Baodong
• Other Names: Tang Xianhua Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2012(2012)
• Publication Date: 2012-02-25
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2012/925092
• Languages: English
• ISSNs: 1110-757X

Abstract

Abstract : LetA be an algebra over a commutative unital ring𝒞 . We say thatA is zero triple product determined if for every𝒞 -moduleX and every trilinear map{ ·, ·, · }, the following holds: if{ x, y, z } = 0 wheneverx y z = 0, then there exists a𝒞 -linear operatorT : A 3 → X such thatx, y, z = T ( x y z ) for allx, y, z ∈ A . If the ordinary triple product in the aforementioned definition is replaced by Jordan triple product, thenA is called zero Jordan triple product determined. This paper mainly shows that matrix algebraM n ( B ), n ≥ 3, where B is any commutative unital algebra even different from the above mentioned commutative unital algebra𝒞, is always zero triple product determined, andM n ( F ), n ≥ 3, where F is any field with chF ≠ 2, is also zero Jordan triple product determined.