Jordan Derivations And Local Derivations Of The Formal Matrix Rings

In ring theory,different matrices have different roles,which the formal matrix ring plays a very important position.In the paper of Tang Gaohua and Zhou Yiqiang about a class of formal matrix rings,which introduces a number of properties in formal matrix ring M_n(R;s)that defined by a central element s in R.This article is inspired by this idea,extending some properties of the formal matrix ring through central element s and draw some relevant conclusions.This master thesis mainly consists of four chapters.Chapter 1,we mainly introduced in this article involves the relevant formal matrix ring M_n(R;s)some of the basic concepts and symbols.Chapter 2,when R is a commutative unitary ring,we obtain the equivalent characterization of the derivative D on the formal matrix ring M_n(R;s):D can be decomposed into the sum of an internal derivative and an induced derivative;when R is a non-commutative unitary ring,we obtain the form that the derivative of M_n(R;s)can be decomposed into s-weakly-derivative.The demonstrative method and result of this part plays a decisive role in Chapters 3 and 4.Chapter 3,the characterization of the Jordan derivative on the formal matrix ring M_n(R;s)and its connection with the derivative are discussed.With the demonstrative method and the result of the first chapter,we obtain that Jordan derivatives of M_n(R;s)is derivatives when R is a commutative 2-torsional union ring,and Jordan derivatives of M_n(R;s)is s-weakly-derivatives when R is a non-commutative 2-torsional union ring.Chapter 4,the relationship between the local derivative and the derivative on the formal matrix ring M_n(R;s)is discussed.We obtain that the 3-local derivatives of the formal matrix ring M_n(R;s)on commutative unitary ring R is the derivatives,and the 3-Local Jordan derivatives is also a derivatives.When9)?3,the n-local derivatives is a derivatives.