共查询到20条相似文献,搜索用时 15 毫秒
1.
Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f( x 1,…, x n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r 1,…, r n ∈ R, a[ F 2( f( r 1,…, r n )), f( r 1,…, r n )] = 0, then one of the following statements hold: 1. a = 0; 2. There exists λ ∈ C such that F( x) = λ x, for all x ∈ R; 3. There exists c ∈ U such that F( x) = cx, for all x ∈ R, with c 2 ∈ C; 4. There exists c ∈ U such that F( x) = xc, for all x ∈ R, with c 2 ∈ C. 相似文献
4.
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f( x1,…, xn) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d( F( f( r)) f( r) ? f( r) G( f( r))) = 0 for all r = ( r1,…, rn) ∈ Rn, then one of the following holds: There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x1,…, xn)2 is central valued on R; There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R; There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C; R satisfies s4 and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R; There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x1,…, xn)2 is central valued on R. 相似文献
5.
Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R2 → R satisfying G( u, u) u = uG( u, u), and G(1, r) = G( r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ( xyx?1y?1) = θ( x)θ( y)θ( x) ?1θ( y) ?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism. 相似文献
6.
Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([ d( x), x] n ) = 0 for all x ∈ R, then char R = 2, d 2 = 0, and δ = α d, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[ d( x), x] n | x ∈ R} in R coincides with the center of R. 相似文献
7.
For a commutative ring R, assume that c is a nonzero element of Z( R) with the property that cZ( R) = {0}. A local ring R is called c- local if Z( R) 2 = {0, c}, Z( R) 3 = {0}, and xZ( R) = {0} implies x ∈ {0, c}. For any finite c-local ring ( R, 𝔪), it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and classification up to isomorphism of all finite commutative c-local rings with order greater than 2 5 are determined. 相似文献
8.
Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f( x 1,…, x n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f( R) the set of all evaluations of the polynomial f( x 1,…, x n ) in R. If [ G( u) u, G( v) v] = 0, for any u, v ∈ f( R), we prove that there exists c ∈ U such that G( x) = cx, for all x ∈ R and one of the following holds: 1. f( x 1,…, x n ) 2 is central valued on R; 2. R satisfies s 4, the standard identity of degree 4. 相似文献
9.
Let R be a prime ring of characteristic different from 2 with Z the center of R and d a nonzero derivation of R. Let k, m, n be fixed positive integers. If ([ d( x k ), x k ] n ) m ∈ Z for all x ∈ R, then R satisfies S 4, the standard identity in 4 variables. 相似文献
10.
Let R be a semiprime ring with Q ml ( R) the maximal left ring of quotients of R. Suppose that T: R → Q ml ( R) is an additive map satisfying T( x 2) = xT( x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q ml ( R) such that T( x) = xa for all x ∈ R. 相似文献
11.
Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I 2) implies a ∈ I or b ∈ I. Let φ: ?( R) → ?( R) ∪ {?} be a function where ?( R) is the set of ideals of R. We call a proper ideal I of R a φ- prime ideal if a, b ∈ R with ab ∈ I ? φ( I) implies a ∈ I or b ∈ I. So taking φ ?( J) = ? (resp., φ 0( J) = 0, φ 2( J) = J 2), a φ ?-prime ideal (resp., φ 0-prime ideal, φ 2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals. 相似文献
12.
Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n( u, w) ≥1 such that ( F( uw) ? bwu) n = 0, then one of the following statements holds: F = 0 and b = 0; R ? M2(K), the ring of 2 × 2 matrices over a field K, b2 = 0, and F(x) = ?bx, for all x ∈ R. 相似文献
13.
Let R be a prime ring with center Z and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f( x) n ∈ Z for all x ∈ L, where n is a fixed positive integer. Then f = 0 unless dim C RC = 4. 相似文献
14.
In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr( a) = P⊕ L, where P ? Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e 2 = e ∈ a n R such that (1 ? e) a n ∈ J( R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/ J( R) is π-regular and idempotents can be lifted modulo J( R). 相似文献
15.
ABSTRACT Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ 1, δ 1] ··· [Θ n , δ n ] an iterated differential operator k-algebra over R such that δ j (Θ i ) ∈ R[Θ 1, δ 1] ··· [Θ i?1, δ i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U( g) the enveloping algebra of g. Then the crossed product of R by U( g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero. 相似文献
16.
Let G be a finite group and let r∈ ?. An r-coloring of G is any mapping χ: G→{1,…, r}. Colorings χ and ψ are equivalent if there exists g∈ G such that χ( xg?1) = ψ( x) for every x∈ G. A coloring χ is symmetric if there exists g∈ G such that χ( gx?1g) = χ( x) for every x∈ G. Let Sr( G) denote the number of symmetric r-colorings of G and sr( G) the number of equivalence classes of symmetric r-colorings of G. We count Sr( G) and sr( G) in the case where G is the dihedral group Dn. 相似文献
17.
ABSTRACTLet n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that ( F( x) F( y)? yx) n = 0 for all x, y∈ L, then char( R) = 2 and R? M2( C), the ring of 2×2 matrices over C. 相似文献
19.
Abstract Let R be a prime ring of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, a ∈ R, S 4( x 1,…, x 4) the standard polynomial in 4 variables. Suppose that, for any x, y ∈ I, a[ d([ x, y]), [ x, y]] = 0. If S 4( I, I, I, I) I ≠ 0, then aI = ad( I) = 0. 相似文献
20.
Abstract Let R be a ring with identity such that R +, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End( R +) = { x ? ax : a ∈ R} and is called an A-ring if Aut( R +) = { x ? ux : u ∈ U( R)}, where U( R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut( R +) there is some natural number n such that α n ∈ { x ? ux : u ∈ U( R)}. We will find all tffr AA-rings with nilradical N( R) ≠ {0} and show that all tffr AA-rings with N( R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings. 相似文献
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