A Study on Derivations in Rings with Engel Conditions

Abstract

Ring theory is a showpiece of mathematical uni cation, bringing together several branches of the subject and creating a powerful machine for the study of problems of considerable historical and mathematical importance. A ring will be de ned as an abstract structure with a commutative addition, and a multiplication which may or may not be commutative. This distinction yields two quite di erent theories: the theory of respectively commutative or non-commutative rings. Rings with derivations are not the kind of subject that undergoes tremendous revolutions. However, this has been studied by many algebraists in the last 50 years, specially the relationships between derivations and the structure of rings. A classical problem of ring theory is to nd combinations of properties that force a ring to be commutative. The present thesis entitled \A Study On Derivations In Rings With Engel Conditions". This exposition comprises two chapters and each chapter is subdivided into various sections. Chapter 1 contains preliminary notions, basic de nitions, examples and some important well-known results related to our study which may be needed for the development of the subject in the next chapter. This chapter is an attempt to make this thesis as self contained as possible. However, the basic knowledge of ring theory has been preassumed. Chapter 2 deals with the study of Engel conditions with derivations in prime and semiprime rings. There is a technique for investigating commutativity of rings is the use of additive mappings like derivations and automorphisms of the ring R. The study of such mappings was initiated by Posner. In [41] Posner proved that if a prime ring R admits a nonzero derivation d such that [d(x); x] 2 Z(R) for all x 2 R, then R is commutative. A number of authors have extended this theorem of Posner. Then in [34] Lee generalized Posner's result. Lee states that if char(R) 6= 2 and [d(x); x] 2 Z for all x in a noncentral Lie ideal of R, then R is commutative. i In this chapter, I have studied the generalization of above results which is proved by Yu Wang [45] from the commutator type to the Engel condition. An additive mapping d : R ! R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x; y 2 R. R is always a prime ring with the center Z(R) . C is its extended centroid and Q is its Martindale quotient ring. For any x; y 2 R, we set [x; y]1 = [x; y] = xy 􀀀 yx and [x; y]n = [[x; y]n􀀀1; y], where n > 1 is an integer. Note that the engel condition is a polynomial [x; y]n = n Pi=0 (􀀀1)i n i !yixyn􀀀i .