Tsiu-Kwen Lee

Generalized derivations of left faithful rings

Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.

Armendariz and Reduced Rings

Abstract A ring R is called Armendariz if, whenever in R[x], a i b j = 0 for all i and j. In this paper, some “relatively maximal” Armendariz subrings of matrix rings are identified, and a necessary and sufficient condition for a trivial extension to be Armendariz is obtained. Consequently, new families of Armendariz rings are presented.

MODULES WHICH ARE INVARIANT UNDER AUTOMORPHISMS OF THEIR INJECTIVE HULLS

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended.

IDENTITIES WITH GENERALIZED DERIVATIONS

Let R be a prime ring with extended centroid C. By a generalized derivation of R we mean an additive map g: R → R such that (xy) g = xgy + xy δ for all x ∈ R, where δ is a derivation of R. In this paper we prove a version of Kharchenkos theorem for generalized derivations and present some results concerning certain identities with generalized derivations.

Left annihilators characterized by GPIs

Let R be a semiprime ring with extended centroid C, U the right Utumi quotient ring of R, S a subring of U containing R and PI, P2 two right ideals of R. In the paper we show that 1S(Pi) = 1S(P2) if and only if Pi and P2 satisfy the same generalized polynomial identities (GPIs) with coefficients in SC, where lS(pi) denotes the left annihilator of pi in S. As a consequence of the result, if p is a right ideal of R such that IR(P) = 0, then p and U satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of R. This paper is motivated by Chuangs paper [3] and Beidars paper [2]. Recall that a ring R is said to be a left faithful ring if, for a E R, aR = 0 implies a = 0. For a left faithful ring R, the right Utumi quotient ring of R can be characterized as a ring U satisfying the following axioms: (1) R is a subring of U. (2) For each a E U, there exists a dense right ideal p of R such that ap C R. (3) If a E U and ap = 0 for some dense right ideal p of R, then a = 0. (4) For any dense right ideal p of R and for any right R-module homomorphism q : PR -RR, there exists a E U such that q(x) = ax for all X E P. Let R be a left faithful ring and p be a dense right ideal of R. We note that p itself is a left faithful ring. Furthermore, p and R have the same right Utumi quotient ring. More precisely, denote by U(R) (U(p) resp.) the right Utumi quotient ring of R (p resp.). Then there exists a ring isomorphism h from U(p) onto U(R) such that h(x) = x for all x E p . In [3] Chuang proved the theorem: Let R be a prime ring, U its right Utumi quotient ring and NR a dense R-submodule of UR. Then N and U satisfy the same generalized polynomial identities (GPIs) with coefficients in U. In this theorem we note that N n R is always a dense right ideal of R. Since N n R and R have the same right Utumi quotient ring, Chuangs theorem just says that R and U satisfy the same GPIs with coefficients in U. Also, in an earlier paper [2] Beidar proved that the same result remains true for semiprime rings. For a semiprime ring R we observe that N n R is a dense right ideal of R for any dense R-submodule NR of UR . Also, lu (N n R), the left annihilator of N n R Received by the editors April 11, 1994 and, in revised form, July 8, 1994; originally communicated to the Proceedings of the AMS by Ken Goodearl. 1991 Mathematics Subject Classification. Primary 1 6R50.

Derivations with Engel conditions on multilinear polynomials

Let R be a prime algebra over a commutative ring K with unity and let f(X1, . . . ,Xn) be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that for all r1, . . . , rn in some nonzero ideal I of R, [ d ( f(r1, . . . , rn) ) , f(r1, . . . , rn) ] k = 0 with k fixed. Then f(X1, . . . , Xn) is central–valued on R except when char R = 2 and R satisfies the standard identity s4 in 4 variables. Throughout this note K will denote a commutative ring with unity and R will denote a prime K–algebra with center Z. By d we always mean a nonzero derivation onR. For x, y ∈ R, set [x, y]0 = x, [x, y]1 = [x, y] = xy−yx and, for k > 1, [x, y]k = [ [x, y]k−1, y ] . A well–known result proved by Posner [10] states that R must be commutative if [ d(x), x ] ∈ Z for all x ∈ R. In [7], the authors generalized Posner’s theorem by showing that a Lie ideal L of R must be contained in Z if char R 6= 2 and [ d(x), x ] ∈ Z for all x ∈ L. As to the case when char R = 2, Lanski [5] obtained the same conclusion except when R satisfies the standard identity s4 in 4 variables. On the other hand, Vukman [11] showed that R is commutative if char R 6= 2 and [ d(x), x ] 2 = 0 for all x ∈ R, or if char R 6= 2, 3 and [ d(x), x ] 2 ∈ Z for all x ∈ R. In a recent paper [6], a full generalization of these results was proved by Lanski. He showed that a Lie ideal L of R is in Z if for some fixed k > 0, [ d(x), x ] k = 0 for all x ∈ L, unless char R = 2 and R satisfies s4. Note that a noncentral Lie ideal of R contains all the commutators [x, y] for x, y in some nonzero ideal of R except when char R = 2 and R satisfies s4. It is natural to consider the situation when [ d(x), x ] k = 0 for all commutators x = [x1, x2], or more generally, when [ d(x), x ] k = 0 for all x = f(x1, . . . , xn) where f(X1, . . . , Xn) is a multilinear polynomial over K. In the present paper, we shall extend Lanski’s theorem by imposing the condition [ d ( f(x1, . . . , xn) ) , f(x1, . . . , xn) ] k = 0 on some nonzero ideal of R. First we dispose of the simplest case when R is the matrix ring Mm(F ) over a field F and d is an inner derivation on R. Received by the editors November 4, 1994 and, in revised form, March 1, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16R50, 16U80.

Derivations with invertible values on a multilinear polynomial

Let R be a semiprime K-algebra with unity, d a nonzero derivation of R, and f(x., xt) a monic multilinear polynomial over K such that d(f(al., at))

Annihilators of power values of derivations in prime rings

0 for some a, , at E R. It is shown that if for every r1, *.., rt in R either d(f(r1, ..., rt)) = 0 or d(f(r1, ..., rt)) is invertible in R, then R is either a division ring D or M2(D) , the ring of 2 x 2 matrices over D, unless f(xl, ... , xt) is a central polynomial for R. Moreover, if R = M2(D), where 2R

A result on derivations

0 and f(xl, ..., xt) is not a central polynomial for D, then d is an inner derivation of R. In [4] Bergen, Herstein, and Lanski proved that if R is a ring with unity and d

Linear identities and commuting maps in rings with involution

0 is a derivation of R such that for every x E R, d(x) = 0 or d(x) is invertible in R, then except for a special case which occurs when 2R = 0, R must be a division ring D or M2(D), the ring of 2 x 2 matrices over a division ring D. In [5] Bergen and Carini gave a generalization of this result to the case of a Lie ideal. More precisely, for the semiprime case they proved: Let R be a semiprime ring with 1, U a noncentral Lie ideal of R such that d(U)