Characterizations of K-Derivations and Bi-Derivations on Lie Ideals of Gamma Rings

Abstract

The present thesis entitled, "CHARACTERIZATIONS OF K-DERIVATIONS AND BI-DERIVATIONS ON LIE IDEALS OF GAMMA RINGS" is the outcome of researches carried out by me under the close supervisions of Dr. Akhil Chandra Paul, Professor, Department of Mathematics, Rajshahi University, Rajshahi.The main goal of this thesis is to characterize k­derivations and then to generalize it. At the begining we introduce the concept of gamma rings and then we have mentioned the extention works on gamma rings, that means k­derivations, jordan k-derivations, Jordan generalized k-derivations, bi­derivations e.t.c. We have also mentioned the mathemeticians who have worked on these fields. In the first chapter we mainly described our works on Jordan k­derivations. Here we have given the definition of k-derivation, Lie ideal, Jordan k-derivation etc. Some examples are also given. Here we define cpa (u, v) for u, v E U, a E r; where U is a Lie ideal of a r-ring M.When M is a 2-torsion free prime r-ring , then we have proved that every Jordan k-derivation is a k-derivation also. In the second chapter, we have discussed about Jordan generalized k­derivation. Here we have define a new additive mapping, which is a generalized form of k-derivation.We have defined it and then defined Jordan generalized k-derivation. For a generalized k-derivation we have defined it with k-derivation. We added some examples. We have defined Jordan generalized k-derivation on Lie ideals. Here we have also defined \Va (u, v), for u, v E U, a E r; where U is a Lie ideal of a r-ring M. At the end of this chapter we have proved that every Jordan generalized k- derivation is a Jordan k-derivation and so is a k-derivation on a Lie ideal of a 2-torsion free prime r-ring M also. We have worked on semi prime r -rings in the third chapter. In this chapter we have characterized semi prime r -rings and proved that every Jordan k-derivation on a Lie ideal U of a 2-torsion free r-ring Mis a k­derivation on U of M if Mis semi prime. The conception of Left centralizer is presented in the forth chapter. We have denoted it by T. Here we have mentioned an additive mapping Ba( u, v) for all u, v E U; a E r.we have also defined Jordan left centralizers. Here we have proved that every Jordan Left centralizer T is a Left centralizer ifM is a 2- torsion free semiprime r-ring. In the fifth chapter, we have discussed Jordan generalized k-derivations on Lie ideals of semiprime r-rings. We have studied Jordan generalized k-derivations earlier. We have worked those on semiprime r-rings. With a special condition we have proved first that every Jordan generalized k­derivation on a Lie ideal U of a 2-torsion free semiprime r-ring M is also a generalized k-derivation on U of M. Then we have proved the same result without any special condition by using the left centralizer. In the sixth chapter, we have studied bi-derivations. We have defined symmetric mapping, bi-derivation, symmetric bi-derivation etc. Here we have also defined trace, which is associated with a bi-derivation. Using different types of conditions, we have proved that either the Lie ideal U is contained in Z(M) or the trace d is zero, if M is a prime r -ring. We have developed these results for a semiprime r-ring also. We have worked on bi-additive mappings on semiprime r-rings in chapter seven. In this chapter we have studied the commutativity of a Lie ideal of a 2-torsion free semiprime r-ring. In the eighth chapter, we have discussed symmetric bi-derivations with symmetric generalized bi-derivations. Here we have worked with two symmetric bi-derivations associated with their respective traces and have found some important results. We have worked on commutativity with symmetric bi-derivations in the nineth chapter. K. K. Dey and A. C. Paul have worked on symmetric bi­derivations. Some of their results are extended here on Lie ideals of prime and semiprime r-rings. At the first stage of this thesis we have discussed about Nobusawa r­rings. In the tenth chapter we have tried to characterize k-derivations on Lie ideals of Nobusawa r-rings. When M is a r-ring, then it is clear that r is also an M-ring. In the basis of this idea we have found ad-derivation on a Lie ideal Q of an M-ring r. In this chapter, we have tried to find out the same types of results on these two categories of derivations on respective Lie ideals of those rings. Also we have worked on d2 and d3. In the eleveth chapter, we have described Left k-derivation. We have defined Jordan left k-derivation also. In this chapter we have used Mas a completely prime r-ring. We have proved that if M is a 2-torsion free completely prime r-ring and U is a Lie ideal of M, then every Jordan left k-derivation on U ofM is also a left k-derivation on U ofM. A complete bibliography which have been helped us to finish my total research is also added at the end of this thesis.