Algebra Colloquium

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. For α = ∑ α_ℓ ℓ in a loop ring RL, define α^♯ = ∑ α_ℓ ℓ^-1 and call α symmetric if α^♯ = α. We find necessary and sufficient conditions under which the symmetric units are closed under multiplication (and hence form a subloop of the loop of units in RL) when R has characteristic two and when R= Z, the ring of rational integers.

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X₁, …,X_t) a multilinear polynomial over K. If g(f (X₁, …,X_t))ⁿ ∈ Z for all x₁, …, x_t ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x₁, …,x_t) is central-valued on R except when R satisfies s₄, the standard identity in four variables.

In this paper, the structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the minimal condition is described, taking as a model the known theory of groups in which normality is a transitive relation.

In this paper, we investigate a general and determinantal representation, and conditions for the existence of a nonzero {2}-inverse X of a given complex matrix A. We introduce a determinantal formula for X, representing its elements in terms of minors of order s = rank(X), 1 ≤ s ≤ r = rank(A), taken from the matrix A and two adequately selected matrices. In accordance with these results, we find restrictions of the adjoint mapping such that the set A{2} is equal to the union of their images. Minors of {2}-inverses are also investigated. Restrictions to the set of {1, 2}-inverses produce the known results from [1–3, 10]. Also, in a partial case, we get known results from [11] relative to the Drazin inverse.

A ring R is called right distributive if its lattice of right ideals is distributive. In this paper, we investigate distributive rings. We prove that if a ring R is right hereditary, then R is right distributive if and only if R is weakly right duo. We also prove that right semiperfect right distributive rings are right quasi-continuous. Finally, it is proved that right perfect distributive rings are quasi-Frobenius. In addition, we add examples to the situations that occur naturally in the process of this paper.

In this paper, we deal with the concept of weakly externally commutative semigroups. We give some basic properties of weakly externally commutative semigroups and describe them as various extensions of commutative semigroups.

The chains of ideals of differentiably simple (non-associative) color algebras and those of their centroids are studied. It is proved that for any finite-dimensional differentiably simple (non-associative) color algebra A and any set D of its color derivations, A is D-simple if and only if its centroid 𝙲(A) is D*-simple, where 𝙲(A) is a unitary ∊-commutative associative color algebra, and D* is the set of the color derivations determined by D.

Let C be a curve, and l, l₀ be lines in the projective three space ℙ³. Consider a projection π_l: ℙ³ ⋯ → l₀ with center l, where l ∩ l₀ = ∅. Restricting π_l to C, we get a morphism π_l|_C: C → l₀ and an extension of fields (π_l|_C)* : k(l₀) ↪ k(C). We study the algebraic structure of the extension and the geometric structure of C. In particular, we study the structure of the Galois group and the number of Galois lines for some special cases.

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

In this paper, we first construct some new infinite dimensional Lie algebras by using the integrable modules of affine Lie algebras. Then we prove that these new Lie algebras are complete. We also prove that the generalized Borel subalgebras and the generalized parabolic subalgebras of these Lie algebras are complete.

We define a degree stable Lie algebra. Since the special type Lie algebra S⁺(2) is degree stable, we find the automorphism group Aut_Lie(S⁺(2)) of the Lie algebra S⁺(2) and prove the Jacobian conjecture of the Lie algebra S⁺(2).

In this paper, we give correspondences between unary algebras, semigroups and congruences on free semigroups. We establish isomorphisms between the complete lattice of varieties of semigroups and the complete lattices of families of varieties of unary algebras, and families of filters of congruences on free semigroups. Similar correspondences between generalized varieties and pseudovarieties of semigroups and corresponding families of algebras and congruences are also established.

This paper studies the finite groups G that have a pair (A,B) of non-trivial proper subgroups satisfying the condition that for every H ⩽ G either A ⩽ H or H ⩽ B.

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab^-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

We study the injective endomorphisms φ of the Baumslag–Solitar group G(n,m)= 〈a,t | t^-1 aⁿt=a^m〉 for n, m ∈ ℤ∖{0} such that n, m ≠ ± 1, and show that φ (a) ∈ ∪ _{i ∈ ℤ∖{0}} (aⁱ)^G. The proof is based on the Bass–Serre theory of graphs of groups.

Let F be the free group of rank 2 with basis {x, y}, and G a metabelian product of some non-trivial abelian groups. If not all the factors of G are torsion groups, it is proved that the verbal subgroup of G in F equals F″. Moreover, all the 2-symmetric words of G are determined by using left Fox derivatives. In addition, we provide an example to illustrate that if all the factors of G are torsion groups, the above results need not be true.