Characterizations of Prime and Semiprime Gamma Rings with Derivations and Lie Ideals

Abstract

Gamma ring was first introduced by N. Nobusawa as a generalization of a classical ring. W. E. Barnes generalized the definition of a gamma ring due to Nobusawa. Presently the gamma ring due to Barnes is known as a gamma ring and gamma ring due to Nobusawa is known as gamma ring in the sense of Nobusawa and is denoted by ����N-ring. It is clear that every ring is a gamma ring and every ����N-ring is also a ����-ring. Actually, W.E.Barnes, J.Luh and S.Kyuno studied the structures of ����-rings and obtained various generalizations analogous to the corresponding parts in ring theory. Afterwards, a number of algebraists have determined a lot of fundamental properties of ����-rings to classify and extend numerous significant results in classical ring theory to ����-ring theory. This thesis, entitled “Characterizations of Prime and Semiprime Gamma Rings with Derivations and Lie Ideals”, aims to characterize prime and semiprime ����-rings with various types of left derivations, derivations, generalized derivations, higher derivations, derivations on Lie ideals and (U,M)-derivations. All the necessary introductory definitions and examples of ����-rings are discussed in considerable details in the introduction chapter. The notions of derivation and Jordan derivation in ����-rings have been introduced by M. Sapanci and A. Nakajima. Afterwards, in the light of some significant results due to Jordan left derivation of a classical ring obtained by K.W.Jun and B.D.Kim, some extensive results of left derivation and Jordan left derivation of a ����-ring were determined by Y.Ceven. In classical ring theory, Joso Vukman proved that if d is a Jordan left derivation of a 2-torsion free semiprime ring R and if there exists a positive integer n such that D(x)n = 0 for all x 2 R, then D = 0. He also proved that for a 2-torsion free and 3-torsion free semiprime ring R admits Jordan derivation D and G : R ! R such that D2(x) = G(x) for all x 2 R, then D = 0. In chapter 1, we extend this result to the ����-ring theory in the case of Jordan left derivations. Then we construct some relevant results to prove that under a suitable condition every nonzero Jordan left derivation d of a 2-torsion free prime ����-ring M induces the commutativity of M, and consequently, d is a left derivation of M. Developing a number of important results on Jordan derivations of semiprime ����- rings, we then prove under a suitable condition, every Jordan derivation of a ����-ring M is a derivation of M, if we consider M as a 2-torsion free (i) semiprime, and (ii) completely semiprime ����-ring, respectively. We examine all these statements in chapter 2 for the clear understanding of the concepts. M. Asci and S. Ceran obtained some commutativity results of prime ����-rings with left derivation. Some commutativity results in prime rings with Jordan higher left derivations were obtained by Kyuoo-Hong Park on Lie ideals and obtained some fruitful results relating this. We work on Jordan higher left derivation on a 2-torsion free prime ����-ring and we show that under a suitable condition, the existence of a nonzero Jordan higher left derivation on a 2-torsion free prime ����-ring M forces M commutative. For the classical ring theories, Herstein, proved a well known result that every Jordan derivation in a 2-torsion free prime ring is a derivation. Bresar proved this result in semiprime rings. Sapanci and Nakajima proved the same result in completely prime ����-rings. C. Haetinger worked on higher derivations in prime rings and extended this result to Lie ideals in a prime ring. We introduce a higher derivation and a Jordan higher derivation in ����-rings. Then we determine some immediate consequences due to Jordan higher derivation of ����-rings to prove under a suitable condition every Jordan higher derivation of a 2-torsion free prime ����-ring M is a higher derivation of M. Y. Ceven and M. A. Ozturk worked on Jordan generalized derivations in ����-rings and they proved that every Jordan generalized derivation on some ����-rings is a generalized derivation. A. Nakajima defined the notion of generalized higher derivations and investigated some elementary relations between generalized higher derivations and higher derivations in the usual sense. They also discussed Jordan generalized higher derivations and Lie derivations on rings. W. Cortes and C. Haetinger proved that every Jordan generalized higher derivations on a ring R is a generalized higher derivation. M. Ferrero and C. Haetinger proved that every Jordan higher derivation of a 2-torsion free semiprime ring is a higher derivation. C. Haetinger extended the above results of prime rings in Lie ideals. By the motivations of above works, we introduce a Jordan generalized higher derivations in ����-rings. We prove that every Jordan generalized higher derivation in a 2-torsion free prime ����-ring with the condition a b c = a b c for all a, b, c 2 M and , 2 ����, is a generalized higher derivation of M. Chapter 3 deals with all these important results elaborately. The relationship between usual derivations and Lie ideals of prime rings has been extensively studied in the last 40 years. In particular, when this relationship involves the action of the derivations on Lie ideals. In 1984, R. Awtar extended a well known result proved by I. N. Herstein to Lie ideals which states that, “every Jordan derivation on a 2-torsion free prime ring is a derivation”. In fact, R. Awtar proved that if U * Z is a square closed Lie ideal of a 2-torsion free prime ring R and d : R ! R is an additive mapping such that d(u2) = d(u)u + ud(u) for all u 2 U, then d(uv) = d(u)v + ud(v) for all u, v 2 U. M. Ashraf and N. Rehman studied on Lie ideals and Jordan left derivations of prime rings. They proved that if d : R ! R is an additive mapping on a 2-torsion free prime ring R satisfying d(u2) = 2ud(u) for all u 2 U, where U is a Lie ideal of R such that u2 2 U for all u 2 U then d(uv) = d(u)v + ud(v) for all u, v 2 U. A. K. Halder and A. C. Paul extended the results of Y. Ceven in Lie ideals. We generalize the Awtar’s result in ����-rings by establishing some necessary results relating to them at the beginning of the chapter 4, we then prove if U is an admissible Lie ideal of a 2-torsion free prime ����-ring M satisfying the condition a b c = a b c for all a, b, c 2 M; , 2 ���� and d : M ! M is a Jordan derivation on U of M, then d(u v) = d(u) v + u d(v) for all u, v 2 U; 2 ���� and if U is a commutative square closed Lie ideal of M, then d(u v) = d(u) v + u d(v) for all u, v 2 U and 2 ����. Accordingly, we then define Jordan higher derivation and higher derivation on Lie ideals of ����-rings and construct some relevant results to prove the previous results analogously in case of Jordan higher derivation on Lie ideal of a ����- ring. Chapter 4 is devoted to a study of these materials in order to bring out the concepts defined clearly. M. Ashraf and N. Rehman considered the question of I. N. Herstein for a Jordan generalized derivation. They showed that in a 2-torsion free ring R which has a commutator right nonzero divisor, every Jordan generalized derivation on R is a generalized derivation on R. In 2000, Nakajima defined a generalized higher derivation and gave some categorical properties. He also treated generalized higher Jordan and Lie derivations. Later, Cortes and Haetinger extended Ashraf’s theorem to generalized higher derivations. They proved that if R is 2-torsion free ring which has a commutator right nonzero divisor, then every Jordan generalized higher derivation on R is a generalized higher derivation on R. Following the notions of Jordan derivation and derivation on Lie ideals of a ����-ring in the previous chapter we then introduce the concepts of a Jordan generalized derivation and generalized derivation on Lie ideals of a ����-ring and we extend and generalized the above mentioned result by these newly introduced concepts. Accordingly, we then define Jordan generalized higher derivation and generalized higher derivation on Lie ideals of a ����-ring and generalized the same result by these concepts. We examine all these statements in chapter 5 for the clear understanding of the concepts. (U,R)-derivations in rings have been introduced by A. K. Faraj, C. Haetinger and A. H. Majeed as a generalization of Jordan derivations on a Lie ideal of a ring. We introduce (U,M)-derivations in ����-rings as a generalization of Jordan derivations on a Lie ideal of a ����-ring. We construct some useful consequences of (U,M)-derivation of a prime ����-ring to prove first that, d(u v) = d(u) v + u d(v) for all u, v 2 U, 2 ����, where U is an admissible Lie ideal of M and d is a (U,M)-derivation of M. We also prove that, if u u 2 U for all u 2 U and 2 ���� then d(u m) = d(u) m + u d(m) for all u 2 U,m 2 M and 2 ����. After introducing (U,M)-derivation in ����-rings, we then introduce the concept of higher (U,M)-derivation in ����-rings. We conclude the chapter 6 by proving the analogous results corresponding to the previous results considering higher (U,M)-derivations of prime ����-rings almost similar way after developing a number of results regarding this newly introduced concept. Following the notion of (U,M)-derivation and higher (U,M)-derivation of a ����- ring in the previous chapter here we introduce the concept of generalized (U,M)- derivation and generalized higher (U,M)-derivation in ����-rings. By establishing some necessary results with generalized (U,M)-derivation, we then prove the analogous results considering generalized (U,M)-derivation of prime ����-rings corresponding to the results of (U,M)-derivation of the previous chapter. A. K. Faraj, C. Haetinger and A. H. Majeed extended Awtar’s theorem to generalized higher (U,R)-derivations by proving that if R is a prime ring, char.(R) 6= 2, U is an admissible Lie ideal of R and F = (fi)i2N is a generalized higher (U,R)-derivations of R, then fn(ur) = P i+j=n fi(u)dj(r) for all u 2 U, r 2 R and n 2 N. Chapter 7 also extends this result in the case of generalized higher (U,M)-derivation of prime ����-rings. Actually, it aims to prove that if U is an admissible Lie ideal of a prime ����-ring M and F = (fi)i2N is a generalized higher (U,M)-derivation of M, then (i) fn(u v) = P i+j=n fi(u) dj(v) for all u, v 2 U, 2 ���� and n 2 N; and also, that (ii) fn(u m) = P i+j=n fi(u) dj(m) for all u 2 U,m 2 M, 2 ���� and n 2 N.