Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

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 ∈ ?.     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

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:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

Range-Inclusive Maps in Rings with Idempotents

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) = λx + 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).     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets 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) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let 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.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D _?1 → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, ?) and Hom(?, 𝒞_𝒲) exact. In this article, we show that C = … → C ^?1 → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) 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,…, r_n) ∈ Rⁿ, then one of the following holds:

1. 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,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. 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;

4. 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;

5. 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,…, x_n)² is central valued on R.

     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

Group Rings Whose Symmetric Units Generate an n-Engel Group

Let F be an infinite field of characteristic different from 2 and G a torsion group. Write 𝒰⁺(FG) for the set of units in the group ring FG that are symmetric with respect to the classical involution induced from the map g ? g ^?1, for all g ∈ G. We classify the groups such that ?𝒰⁺(FG)? is n-Engel.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

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?].