Derivations In Prime Rings And Jordan Maps Of Triangular Algebras

Theory of derivations in rings with generalization is an important topic in theory of rings. It plays a significant role to depict structure of rings. There are many problems and applications in this field to be found and studied. One of aims of this dissertation is to research derivations and generalized derivations in prime rings.In 1957, Posner firstly investigated derivations in prime rings. He proved that a prime ring with a centralizing nonzero derivation is commutative. Posner's results stimu-lated investigations on derivations and other maps in rings. A number of generalizations of Posner's theorem grew up from then on. In 1969, Martindale defined a kind of quotient ring, called Martindale quotient ring, and extended theory of PI to GPI. Theory of quo-tient rings and GPI provide standard tools for study of derivations. In 1978, Kharchenko realized transformation from differecial polynomial identities to generalized polynomial identities, which is a powerful tool to analyse properties of derivations in rings. A large numbers of papers on derivations in prime rings were published from then on. In 1988, Lanski generalized Posner's theorem applying theory of differential identities. He re-garded centralizing condition as a special differential identity, which was a breakthrough method and breaked a new path for study on derivations in rings.Posner's theorem has been generalized by a number of authors in several ways. There were two main ways to do it:One was making identities including derivations complicated through making use of Engel condition, adding powers and multipliers; The other was specializing sets on which identities hold through considering the whole ring, ideals, one-sided ideals, Lie ideals, sets of commutators on a special subset and set consisting of values of a multilinear polynomial and so on.Chapter 3 is devoted to derivations satisfying certain differential identities.Theorem 3.1.1 Let R be a prime ring of characteristic different from 2 with center Z, I a nonzero ideal of R, and d a nonzero derivation of R such that [d(χκ),χκ]n∈Z for allχχI,whereκ, n are fixed positive integers. Then R satisfies s4, the standard identity in 4 variables.Theorem3.1.1 generalizes a result of Carini and Filippis in 2000.Theorem 3.2.1 Let R be a prime ring. L a noncentral Lie ideal of R, and d a nonzero derivation of R. Suppose thatχs[d(χ),χ]κχt=0 for all x E L, where s,κ,t≥0 are fixed integers. Then char(R)= 2. Moreover, if either s=0 or t=0, then R C M2(F) for a field F.When s=t=0, Theorem3.2.1 gives a result of Lanski in 1993.We study annihilator conditions with generalized derivations on a multilinear poly-nomial in Chapter 4.The notion of derivation has been generalized in several directions to extend study on derivations. In 1991, Bresar introduced generalized derivations. Let g:Ryou→R be an additive map of a ring R. If there exists a derivation d of R such that g(χy)= g(χ)y+χd(y) for allχ.y∈R. then g is called a generalized derivation of R. Hvala first extended some results on derivations to generalized derivations in 1998. A number of works on generalization of derivations have been done by many authors from then on.Theorem 4.2.1 Let R be a prime ring with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, fαmultilinear polynomial over C, a E R and I a nonzero right ideal of R such thatαI≠0. Suppose that a[g(f(r1,…,rn)),f(r1,…,rn)]= 0 for all ri E I. Then either g(χ)=α1χfor someα1∈E U andμ∈C such that(α1-γ)I=0, or there exists an idempotent element e E soc(RC) such that IC= eRC and one of following holds:(i) f is central valued in eRe; (ii) g(χ)= bχ+χc, where b,c∈U with (c-b-α)e=0 for some a E C and f2 is central valued in eRe;(iii) dimceRCe= 4 and char(R)=2.Theorem4.2.1 unifies and improves two results due to Filippis in 2007 and 2008.Theorem 4.3.1 Let R be a prime ring with the right Utumi quotient ring U, gαgeneralized derivation of R, L a noncentral Lie ideal of R, and 0≠α∈R. Suppose thatαus(g(u))nut=0 for all u∈L, where s,t≥0 and n> 0 are fixed integers. Then s=0 implies that there exists b∈U with ab=0 such that g(χ)=bx for allχ∈R and s>0 implies that g=0, unless R satisfies S4.Theorem4.3.1 improves some results obtained by Dhara and Sharma in 2007 and Dhara and Filippis in 2009.Chapter 5 is focused on additivity of Jordan maps and cocommuting maps on trian-gular algebras.Let A and B be two algebras over a commutative unitary ring K and X be an (A,B)-bimodule. Then the set of formal matricesTri(A,X,B)={(aχb)|α∈A,χ∈X,b∈B} is a K-algebra under the usual matrix addition and formal matrix multiplication, called a triangular algebra.The upper triangular matrix algebras and nest algebras are two standard examples of triangular algebras.Since the late 1940's, the additivity of multiplicative maps has been examined by many authors. In 1969, Martindale gives a sufficient condition for the additivity of a mul-tiplicative maps. Besides the usual multiplication, Jordan multiplication, Jordan triple multiplication, Lie multiplication and multiplicative derivations are also considered. How-ever, Martindale's result is the most fundamental one in mass of results on derivations. His method has been refered by many authors.Let R, S be two rings. M:R→R' and M*:R'→R be two maps. (M, M*) is called a Jordan map of R x R' if for allα∈R,χ∈R', where o denotes Jordan multiplication. Theorem 5.2.1 Let T=Tri(A,X,B), where one of A and B is 2-torsion free,X is both faithful A-module and faithful B-module such that AmB=0, a∈X implies m=0. For any ring R', if (M, M*) is a Jordan map of T×R' such that both M and M* are surjective, Then both M and M* are additive.Theorem5.2.1 improves the results obtained by Ji in 2007 and Wang in 2011, respec-tively.Let A be an algebra over a commutative unitary ring K, F and G be two maps from A into itself. If F(a)a=aG(a),a∈A then F and G are called co-commuting on A. In particular, F is commuting if F with itself is co-commuting; F is skew commuting if F and-F are co-commuting. A pair of co-commuting maps F and G is called proper if for all a∈A, whereχ∈A andμ:A→Z(A).Theorem 5.3.1 Let T=Tri(A,X,B), where A and B are both K-algebras with identity, X is both faithful A-module and faithful. Then every pair of co-commuting maps on T is proper if the following three conditions hold:(ⅰ) Z(B)=πB(Z(T)) or A=[A,A];(ⅱ) Z(A)=πA(Z(T)) or B=[B,B];(ⅲ) There exists m0∈X such that Z(T)={a(?)b|a∈Z(A), b∈Z(B),am0= m0b}.Theorem5.3.1 improves many results on nest algebra and Bresar's results.