Chen-Lian Chuang

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x1, …,xn) be a polynomial overC in noncommuting variablesx1, …,xn. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a1, …,an):a1, …,an ∈I}. Then eitherp(x1, …,xn) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.

On invariant additive subgroups

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)−1 ⊆A for allt ∈R such thatt2=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C2, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C2 andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα2Z ⊆ {β2:β ∈Z}.

Jacobson Radicals of Ore Extensions of Derivation Type

Let D be a sequence of derivations of a ring R. We form the Ore extension R[X; D]. Let 𝒩(R) denote the nil radical of R and 𝒥(R[X; D]), the Jacobson radical of R[X; D]. We show that 𝒩(R) = 0 implies 𝒥(R[X; D]) = 0 under one of the following conditions: (1) R is a PI-ring. (2) One of derivations in D is quasi-algebraic. (3) R satisfies the ascending chain condition on right annihilators.

Extended Jacobson Density Theorem for Rings With Skew Derivations

Let A be a ring with a simple right module M and D = End(MA). Let {δ1, δ2,…, δn} be a reduced set of skew derivations, where each δi is a σi-derivation of A satisfying σiδj = δjσi and σiσj = σjσi for all i, j. Let Δ1,…, Δm be distinct words of the form , where each is < p in the case of char(D) = p > 0. Let α1,…, αℓ be mutually M-independent automorphisms of A. Then for any D-independent elements x1, x2,…, xk ∈ M and any elements zijt ∈ M, there exists a ∈ A such that zijt = xiaΔjαt for all i = 1, 2,…, k, j = 1, 2,…, m, t = 1,…, ℓ.

Algebraic q-skew derivations

Abstract Let R be a prime ring, C its extended centroid and R F (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension Q [ X ; σ , δ ] . We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression ϕ ( x ) : = ∑ i = 0 n a i δ i ( x ) , where a i ∈ R F and a n ≠ 0 . (1) If dim C ϕ ( R ) C ∞ then either R is a GPI-ring or ϕ ( x ) = 0 for all x ∈ R F . (2) If ϕ ( R ) ⊆ C then either R is commutative or ϕ ( x ) = 0 for all x ∈ R F .

On a conjecture by Herstein

Abstract 1. (1) Assume that R has no nonzero nil two-sided ideals. Let K ⩾ l be a fixed positive integer. The following two results are shown: (a) If, to each pair of elements x, y of R, there correspond positive integers m = m(x, y), n=n(x, y) ⩾ 1 such that […[[xm, yn], yn] …, yn]=0, k-times then R is commutative, (b) If a ϵ R is such that, to each y of R, there corresponds a positive integer n = n(a, y) ⩾ 1 such that […[[a, yn], yn] …, yn]=0, k-times then a is in the center of R. 2. (2) Assume that R has no nonzero nil one-sided ideals. Then the same results as in (a) and (b) above are shown under the weaker assumptions that the k in (a) depends on the pair of elements x, y ϵ R and that the k in (b) depends on both a and y.

A NOTE ON CERTAIN SUBGROUPS OF PRIME RINGS WITH DERIVATIONS

ABSTRACT Let be a noncommutative prime ring of characteristic not and a nonzero derivation of . Chebotar and Lee proved that the additive subgroup of generated by the subset contains a noncentral Lie ideal of if . In this note we show that the same conclusion holds even if .

Algebraic derivations with constants satisfying a polynomial identity

We prove that if a semiprime ringR possesses a derivation which is integral over its extended centroidC and whose constants satisfy a polynomial identity, thenR itself is a PI-ring. This answers affirmatively a problem raised by M. Smith in 1975 and recently again by Bergen and Grzeszczuk [4].

Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U s (R)[X; D], where U s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U s (R) such that δ(α) = 0 for all δ ∈ D. These extend the known results for free algebras.

Derivations and Skew Polynomial Rings

Let R be a prime ring and d a derivation of R. In the ring of additive endomorphisms of the Abelian group (R, +), let S be the subring generated by a L d m , where a ∈ R and m ≥ 0 and where a L :x ∈ R ↦ ax ∈ R for a ∈ R. We compute the prime radical and minimal prime ideals of S via the skew polynomial ring R[x; d] by the surjective ring homomorphism We compute explicitly the kernel 𝒜 of ϕ, the prime radical 𝒫 over 𝒜 and minimal prime ideals over 𝒜 (Theorem 2). We obtain a necessary and sufficient condition for S to be simple, prime or semiprime (Corollary 3). As an application, let d be nilpotent. We show that the d-extension of R defined in Grzeszczuk (1992) is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14).