共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
Let ? be a prime ring of characteristic different from 2, 𝒬 r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [ F( r), r] kr ? r[ G( r), r] k = 0 for all r ∈ ?, then there exist a ∈ 𝒬 r and λ ∈ 𝒞 such that F( x) = xa and G( x) = ( a + λ) x, for all x ∈ ?. 相似文献
3.
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
4.
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
5.
We study algebra classes and divisor classes on a normal affine surface of the form z 2 = f( x, y). The affine coordinate ring is T = k[ x, y, z]/( z 2 ? f), and if R = k[ x, y][ f ?1] and S = R[ z]/( z 2 ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H 1( G, Cl( T)) → H 1( G, Pic( S)) factors through the relative Brauer group B( S/ R) and that all of the maps are onto. Sufficient conditions are given for H 1( G, Cl( T)) to be isomorphic to B( S/ R). The groups and maps are computed for several examples. 相似文献
6.
Let F = ? x, y? be a free group. It is known that the commutator [ x, y ?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [ x, y ?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [ x, y ?1] ≡ u, where u ∈ [ F′, ? x?]. 相似文献
7.
Let R be a PID. We construct and classify all coordinates of R[ x, y] of the form p 2 y + Q 2( p 1 x + Q 1( y)) with p 1, p 2 ∈ qt( R) and Q 1, Q 2 ∈ qt( R)[ y]. From this construction (with R = K[ z]) we obtain nontame automorphisms σ of K[ x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms. 相似文献
8.
Let 𝒜 be a ring, let ? be an 𝒜-bimodule, and let 𝒞 be the center of ?. A map F:𝒜 → ? is said to be range-inclusive if [ F( x), 𝒜] ? [ x, ?] for every x ∈ 𝒜. We show that if 𝒜 contains idempotents satisfying certain technical conditions (which we call wide idempotents), then every range-inclusive additive map F:𝒜 → ? is of the form F( x) = λ x + μ( x) for some λ ∈ 𝒞 and μ:𝒜 → 𝒞. As a corollary we show that if 𝒜 is a prime ring containing an idempotent different from 0 and 1, then every range-inclusive additive map from 𝒜 into itself is commuting (i.e., [ F( x), x] = 0 for every x ∈ 𝒜). 相似文献
9.
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
10.
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
11.
Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [ g( r k ), r k ] n = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g( x) = xc and I( c ? α) = 0 for a suitable α ∈ C. In particular we have that g( x) = α x, for all x ∈ I. 相似文献
12.
Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([ x, [ y, z]]) = [ f( x), [ f( y), f( z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms. 相似文献
13.
Let R be a prime ring and set [ x, y] 1 = [ x, y] = xy ? yx for ${x,y\in R}$ and inductively [ x, y] k = [[ x, y] k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [ δ( x), x] k = 0 for all ${x \in L}$ , where k is a fixed positive integer, then char R = 2 and ${R\subseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne. 相似文献
14.
Abstract Let A be a commutative ring with identity, let X, Y be indeterminates and let F( X, Y), G( X, Y) ∈ A[ X, Y] be homogeneous. Then the pair F( X, Y), G( X, Y) is said to be radical preserving with respect to A if Rad(( F( x, y), G( x, y)) R) = Rad(( x, y) R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht( Q/ P) = 2. 相似文献
15.
Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′( R), is a graph with vertex set W*( R), where W*( R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ? Rb and b ? Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′( R) is a unicyclic graph if and only if R ? ? 2 × ? 4, ? 3 × ? 3, ? 2 × ? 2[ x]/( x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′( R) is a unicyclic graph if and only if R ? ? 2 × ? 4, ? 3 × ? 3, ? 2 × ? 2[ x]/( x 2), ? 2[ x, y]/( x, y) 2, ? 4[ x]/(2 x, x 2). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′( R)) ∈ {3, 4, ∞}. 相似文献
16.
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
17.
In this note we study radicals of skew polynomial ring R[ x; α] and skew Laurent polynomial ring R[ x, x ?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J( R[ x; α]) = N 0( R[ x; α]) = Ni? *( R)[ x; α] and J( R[ x, x ?1; α]) = N 0( R[ x, x ?1; α]) = Ni? *( R)[ x, x ?1; α]. 相似文献
18.
Abstract Let d 1 : k[ X] → k[ X] and d 2 : k[ Y] → k[ Y] be k-derivations, where k[ X] ? k[ x 1,…, x n ], k[ Y] ? k[ y 1,…, y m ] are polynomial algebras over a field k of characteristic zero. Denote by d 1 ⊕ d 2 the unique k-derivation of k[ X, Y] such that d| k[X] = d 1 and d| k[Y] = d 2. We prove that if d 1 and d 2 are positively homogeneous and if d 1 has no nontrivial Darboux polynomials, then every Darboux polynomial of d 1 ⊕ d 2 belongs to k[ Y] and is a Darboux polynomial of d 2. We prove a similar fact for the algebra of constants of d 1 ⊕ d 2 and present several applications of our results. 相似文献
19.
Abstract Given a contravariant functor F : 𝒞 → 𝒮 ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair ( X, x) where X is an object of 𝒞 and x ∈ F( X) if for any object Y of 𝒞 and any y ∈ F( Y), there is a morphism f : Y → X such that F( f)( x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F( f)( x) = x is an automorphism of X, we say that F is minimally generated by ( X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ? xt (?, M) : ? → 𝒮 ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules. 相似文献
20.
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
|