A Class Of Zero Product And Jordan Zero Product Determined Algebras

In recent years, many scholars both at home and abroad have made extensive and thorough research on linear preserver problem. Many scholars applied characters of bilinear map on preserve zero product and Jordan zero product, and introduce the definition of zero product determined. Based on existing resulting. in this paper we mainly discuss zero product and Jordan zero product determined algebra, then introduce application of linear preserve. This paper is divided into3chapters.In chapter l,we mainly introduce some notations, basic definitions and main theorems which will be used in the whole paper. We firstly give some notations, and give the definition of zero product determined.In chapter2, we discuss homogeneous bijective Φ:M2(R)â†'M2(R) on M2(R). In this paper we prove that a homogeneous bijective Φ:M2(R)â†'M2(R) which satisfies Φ([A B]T)=[Φ(A) Φ(B)]T for all A, B∈M2(R) if and only if there exists an orthogonal matrix N∈M2(R) such that v(A)=NANT.In chapter3, we show that any algebra generated by idempotents over a commu-tate ring is determined by zero product and zero Jordan product. As an application, we give a characterization of homomorphisms, Jordan homomorphisms, Lie homo-morphisms, derivations, Jordan derivations and Lie derivations on such algebras.