Symmetric Martindale Quotient Ring Research Articles

Generalized derivations with nilpotent, power-central, and invertible values in prime and semiprime rings

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, and fixed integers. Let be a non-zero multilinear polynomial over C in t non-commuting variables, be any subset of R and . We prove the following results:If R is prime and for all , then is central valued on R.If R is prime and , for all , then is power central valued on R, unless .If R is semiprime and for all , then , for any and , that is there exists a central idempotent element such that , d vanishes identically on eQ and is central valued on .If R is semiprime and is zero or invertible in R, for all , then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when , for any and .If R is prime and I is a non-zero right ideal of R such that and , for all , then is an identity on I.Let R be prime and I a non-zero right ideal of R such that and , for all . If there exists such that , then either is power central valued on R or is an identity on I, unless .

Minimal quasi-algebraic relations of skew derivations in characteristic p ≥ 2

Let R be a prime ring of charR=p≥2. The monic minimal quasi-algebraic relation of an ordinary derivation D of R takes the form Dps+∑i=0s−1αiDpi=ad(a), where αi are in the extended centroid of R and where a is in the symmetric Martindale quotient ring of R. Among other things, we extend this result to an arbitrary skew derivation D. This is explained in details in §0 below.

A note on generalized derivations on prime rings

Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring Q_s(R). In this paper we prove the following result. Let F: R rightarrow R be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that F(x)x=xh(x) for all xin R. Then either R is commutative or F(x)=xp and h(x)=px where pin Q_{s}(R).

Symmetric Martindale Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and D a sequence of derivations of R with length ≥2. Let be the subring consisting of those q in the symmetric Martindale quotient ring of R such that qI ∪ Iq ⊆ R for a nonzero D-invariant ideal I of R. It is shown here that the symmetric Martindale quotient ring of the Ore extension R[X, D] is the Ore extension . Our proof depends on an interesting combinatoric result on words.

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X n h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q 1 ⊕ Q 2 ⊕ Q 3 such that Q 1 is a ring satisfying S 2n−2, the standard identity of degree 2n − 2, Q 2 ≅ M n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x q = x for all x ∈ E, and Q 3 is a both faithful S 2n−2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.

On Some Functional Equations in Rings

In this article we prove the following result. Let n ≥ 1 be some fixed integer, and let R be a prime ring with 2n ≤ char(R) ≠ 2. Suppose there exists an additive mapping T: R → R satisfying the relation for all x ∈ R. In this case, T is of the form 4T(x) = qx + xq for all x ∈ R, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.

Derivations modulo elementary operators

Let R be a prime ring with extended centroid C and symmetric Martindale quotient ring Q s ( R ) . Suppose that Q s ( R ) contains a nontrivial idempotent e such that e R + R e ⊆ R . Let ϕ : R × R → R C + C be the bi-additive map ( x , y ) ↦ G ( x ) y + x H ( y ) + ∑ i a i x b i y c i , where G , H : R → R are additive maps and where a i , b i , c i ∈ R C + C are fixed. Suppose that ϕ is zero-product preserving, that is, ϕ ( x , y ) = 0 for x , y ∈ R with x y = 0 . Then there exists a derivation δ : R → Q s ( R C ) such that both G and H are equal to δ plus elementary operators. Moreover, there is an additive map F : R → Q s ( R C ) such that ϕ ( x , y ) = F ( x y ) for all x , y ∈ R . The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems.

Kernel Inclusions of Algebraic Automorphisms

Let R be a prime ring with left Martindale quotient ring R ℱ and symmetric Martindale quotient ring Q. Define, for an automorphism σ of R, R (σ) = {x ∈ R∣x σ = x}. Let σ and τ be automorphisms of R, and assume that σ is left R ℱ -algebraic. We show that R (σ) ⊆ R (τ) if and only if x τ = vx σ i v −1 for all x ∈ R, where i is an integer and where v is in the centralizer of R (σ) in Q.

AN IDENTITY WITH GENERALIZED DERIVATIONS

Let R be a prime ring that is not commutative and such that R ≇ M2( GF (2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(xm)xn= xnG(xm) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or xmand xnare C-dependent for all x ∈ R. We also consider the situation for the semiprime case.

Finiteness properties of differential polynomials

Let R be a prime ring with extended centroid C and let ϕ ( X i Δ j ) be a reduced differential polynomial with coefficients in Q , the symmetric Martindale quotient ring of R , and with zero constant term. Let A ϕ = { ϕ ( x i Δ j ) | x i ∈ R } and B ϕ = { ϕ ( x ij ) | x ij ∈ R } . We prove that the finiteness of A ϕ and the finite-dimensionality of the C -span of A ϕ are equivalent to that of B ϕ and that of the C -span of B ϕ , respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials. Let δ and d be two derivations of R , L a Lie ideal of R and ρ a right ideal of R . As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of d ( ρ ) , d ( L ) and δ d ( L ) and for the finite-dimensionality of the C -spans of d ( ρ ) , d ( L ) and δ d ( L ) .

The kernel and range inclusions of integral derivations in semiprime rings

Let R be a semiprime ring with extended centroid C and with symmetric Martindale quotient ring Q. For a derivation δ of R and an ideal I of R, define Ker I ( δ ) = def . { r ∈ I | δ ( r ) = 0 } and R I ( δ ) = def . δ ( I ) . Let δ , δ ′ be derivations of R such that δ is C-integral and has the associated X-inner derivation ad ( a ) , where a ∈ Q . The main results of this paper (Theorem 3.5) are the following two equivalences: (1) Ker I ( δ ) ⊆ Ker R ( δ ′ ) for an essential ideal I of R if and only if δ ′ = ∑ i ⩾ 1 μ i δ i + ad ( b ) for some μ i ∈ C and some b ∈ C [ a ] . (2) R I ( δ ′ ) ⊆ R R ( δ ) for an essential ideal I of R if and only if δ ′ = ∑ i ⩾ 1 μ i δ i + ad ( b ) for some μ i ∈ C with ∑ i ⩾ 1 ( − 1 ) i δ i ( μ i ) = 0 and some b ∈ C [ a ] .

On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R → R be an additive mapping satisfying the relation T(x3) = T(x)x2 − xT(x)x + x2T(x) for all x ∈ R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.

Ore extensions which are GPI-rings

Let R be a prime ring and δ a σ-derivation of R, where σ is an automorphism of R. It is proved that the skew polynomial ring $$R[t; \sigma, \delta]$$ is a GPI-ring (PI-ring resp.) if and only if R is a GPI-ring (PI-ring resp.), δ is quasi-algebraic, and σ is quasi-inner. If $$R[t; \sigma, \delta]$$ is a GPI-ring then soc $$(Q[t;\sigma,\delta]\tilde C) = \Big({\rm soc}(Q)[t;\sigma,\delta]\Big)\tilde C$$ , where Q is the symmetric Martindale quotient ring of R and where $$\tilde C$$ denotes the extended centroid of $$Q[t; \sigma, \delta]$$ . If $$R[t; \sigma, \delta]$$ is a PI-ring, its PI-degree is determined as follows: $$(1) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\delta)$$ if δ is X-outer, and $$(2) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\sigma)$$ if δ is X-inner.

Generalized Differential Identities of (Semi–)Prime Rings

Let R be a semiprime ring with characteristic p ≥ 0 and RF be its left Martindale quotient ring. If \( \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} \) is a reduced generalized differential identity for an essential ideal of R, then ϕ(Zije(Δj)) is a generalized polynomial identity for RF, where e(Δj) are idempotents in the extended centroid of R determined by Δj. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If \( \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} \) is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ϕ(Zij) is a generalized polynomial identity for [R,R]. Moreover, if \( \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} \) is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ϕ(Zij is a generalized polynomial identity for Q.

On certain equations in rings

In this paper we prove the following result: LetRbe a 2-torsion free semiprime ring. Suppose there exists an additive mappingT:R→Rsuch thatT(xyx)=T(x)yx−xT(y)x+xyT(x)holds for all pairsx, y∈R. ThenTis of the form2T(x)=qx+xq, whereqis a fixed element in the symmetric Martindale ring of quotients ofR.

Generalized skew derivations characterized by acting on zero products

Let A be a prime ring such that its symmetric Martindale quotient ring contains a nontrivial idempotent. Then generalized skew derivations of A are characterized by acting on zero products. Precisely, if g;-:A ! A are additive maps such that ae(x)g(y) + -(x)y = 0 for all x;y 2 A with xy = 0 where ae is an automorphism of A, then both g and - are characterized as specific generalized ae-derivations on a nonzero ideal of A. 1. Results Let B be a ring with a subring A. An additive map -:A ! B is called a derivation if -(xy) = -(x)y + x-(y) for all x;y 2 A. In a recent paper Jing, Lu and Li proved the following result (6, Theorem 6): Let B be a standard operator algebra in a Banach space X containing the identity operator I, and -:B ! B be a linear map such that -(AB) = -(A)B + A-(B) for any pair A;B 2 B with AB = 0. Then -(AB) = -(A)B + A-(B) i A-(I)B for all A;B 2 B. Moreover, if in addition -(I) = 0, then - is a derivation. The result says that an additive map on a standard operator algebra is almost a derivation if it satisfies the expansion formula of derivations on pair elements with zero product. Since standard operator algebras involve many idempotents, from this point of view Chebotar, Ke and P.-H. Lee studied maps acting on zero products in the context of prime rings (2). To give its precise statement we first fix some notation. Throughout, unless specially stated, A always denotes a prime ring with center Z, extended centroid C and symmetric Martindale quotient ring Q. Moreover, let

Hopf module duality applied to X-outer Galois theory

We work over a fixed ground field k in arbitrary characteristic. Let R be a prime algebra, and let Q denote the symmetric Martindale quotient ring of R. We know from [10, Section 6.4] that to study group actions on R, it is fundamental to consider the extended actions on Q, and those actions which are X-outer, or outer as automorphisms of Q, have nice properties, which could be generalized for X-outer (in some appropriate sense) Hopf algebra actions. To recall Kharchenko’s Galois-type correspondence theorem, let G be a finite group of X-outer automorphisms of R. It was proved in [2, Theorem 3.10.2], [11, Theorem B] that

The rings of quotients of free algebras

It is proved that a symmetric Utumi ring of quotients, U, of a free associative (noncommutative) algebra F(X) with unity coincides with the algebra itself, U=F(X). From this, we obtain a similar statement concerning a symmetric Martindale ring of quotients, Q(F(X))=F(X), which is well known. In addition, it is shown that a left Martindale ring of quotients, F(X)F, of a free algebra is a prime algebra and, moreover, every homogeneous element in a free algebra has the right inverse in F(X)F but does not have the left one (unless, of course, r belongs to an underlying field). Since a left Utumi ring of quotients and a left Martindale ring of quotients for a free algebra both appear prime, an interesting question arises as to whether or not they coincide.

Derivation of prime rings of positive characteristic

Let L be a finite-dimensional differential Lie algebra acting on a prime ring R and let the inner part {ie49-1} of L be quasi-Frobenius. Then a constant ring RL is prime iff {ie49-2} is a differentially simple ring. A ring of constants is semiprime iff {ie49-3} is a direct sum of differentially simple rings, and the prime dimension of a constant ring is equal to the number of differentially simple summands {ie49-4}. The Galois closure of L is obtained from L by adding all the inner derivations of a symmetric Martindale quotient ring which agree with elements from {ie49-5}.

Delta Ideals of Lie Color Algebras

Let L = ⊕g ∈ GLg be a Lie color algebra (possibly restricted) over the field K and graded by the finite abelian group G. If Δ∞(L) = {l ∈ L |dimK[l, L] is countable}, then Δ∞(L) is the (restricted) Lie color ideal of L generated by all (restricted) countable-dimensional Lie color ideals of L. We use Δ∞(L) to examine the symmetric Martindale quotient ring of the enveloping algebra U(L) (or the restricted enveloping algebra when char K = p > 0). Specifically, we prove THEOREM. If Δ∞(L) = 0, thenU(L) is symmetrically closed. We also examine the Lie color ideal Δ(L) = {l ∈ L |dimK[l, L] is finite} and the possibly smaller ideal ΔL, which is the join of all finite-dimensional Lie color ideals of L. Note that Δ(L) = ΔL when char K = p > 0, but that Δ(L) can be considerably larger than ΔL, when char K = 0. Nevertheless, we prove THEOREM. [Δ(L), Δ(L)] ⊆ ΔL. We remark that these results are new and of interest even when L is an ordinary or super Lie algebra. In fact, we consider Lie color algebras here only because we can obtain the more general facts with little additional work.