A Generalization of a Lemma of Sullivan

Consider an irreducible polynomial of the form f(X) = X ^p  ? aX ? b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form α ⁿ , where 0 ≤ n ≤ p ² ? 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(α ⁿ ) for n < p ^r can be reduced to computing the traces Tr(α ^m ) for all m ≤ r(p ? 1).     

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.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

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.     

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).     

On Factorizations of Subformations of One-Generated Saturated Finite Varieties

The following is proved: If 𝔐? is the noncancellable product of the formations 𝔐 and ? and 𝔐? ? 𝔉 for some one-generated saturated finite variety 𝔉, then 𝔐 is soluble.     

Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

On Factorizations of Baer-Local Formations

Abstract

It is proved that if the product ? of formations 𝔐 and ? ≠ ? is either a Baer-local formation or an ω-local formation and if ? ? 𝔉 for some one-generated ω-local formation 𝔉, then 𝔐 is also an ω-local formation.     

Invertibility of Matrices Over Subrings

Given rings R ? S, consider the division closure 𝒟(R, S) and the rational closure ?(R, S) of R in S. If S is commutative, then 𝒟(R, S) = ?(R, S) = RT ^?1, where T = {t ∈ R | t ^?1 ∈ S}. We show that this is also true if we assume only that R is commutative.     

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.     

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N _G (K) ∩ K ^x  ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P ₀ ? P ₁ ? ··· ? P _n  = P is called a maximal chain of P provided that |P _i : P _i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P ₀ ? P ₁ ? ··· ? P _i  ? ··· ? P _n  = P such that P _i  ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.     

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).     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

Ideals,Natural Classes,and Functors

For any ring R, the set 𝒩(R) of all natural classes of R-modules is a complete Boolean lattice, which is a direct sum of two convex and complete Boolean sublattices 𝒩(R) = 𝒩 _t (R) ⊕ 𝒩 _f (R), where the last summand is the set of all nonsingular natural classes. The ring R contains a unique lattice of ideals 𝒥(R) which is lattice isomorphic to 𝒩 _f (R). The present note develops the analogue of all of the above for an arbitrary R-module M, so that in the special case when M _R  = R _R , the known lattice isomorphism 𝒥(R) ? 𝒩 _f (R) is recovered.     

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?].     

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.     

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}.