Journal of Algebra and Its Applications

We classify the orbit closures in the variety of complex, three-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra.

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth gr(G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with gr(G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr(G(R)) = 3 or 4.

One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x^-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

Let (R, 𝔪) be a Noetherian local ring with dim R = d and M an R-module with dim M < d. We prove in this paper that the idealization R ⋉ M of M over R is a pseudo Buchsbaum ring if and only if so is R.

The dimensions of the symmetry classes of polynomials with respect to a certain cyclic subgroup of S_m generated by an m-cycle are explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed in terms of the Euler ϕ-function and the Möbius function for some special cases.

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].

A p-group is called a C(p^w)-group if the normal closure of every non-normal cyclic subgroup has index at most p^w. In this paper, we prove that the order of a non-Dedekind C(p^w)-group cannot exceed p^4w+4 when p > 2 and G′ ≤ H^G for every non-normal cyclic subgroup H. We also completely classify non-Dedekind C(p²)-groups for p > 2.

Let be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. We write

Let G be a non-abelian group and N(G) be the set of conjugacy class sizes of G. In 1980s, Thompson posed the following conjecture: if M is a finite non-abelian simple group, G is a finite group with Z(G) = 1 and N(G) = N(M), then M ≅ G. Here, we prove Thompson's conjecture holds for all almost sporadic simple groups.

Let p be an odd prime, D_2p be the dihedral group of order 2p, and F₂ be the finite field with two elements. If * denotes the canonical involution of the group algebra F₂D_2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group , generated by the bicyclic units of the group algebra F₂D_2p. Further, we obtain the structure of the unit group and the unitary subgroup , and we prove that both and are normal subgroups of .

We show that the repetitive higher cluster category of type A_n, defined as the orbit category , is equivalent to a category defined on a subset of diagonals in a regular polygon. This generalizes the construction of Caldero–Chapoton–Schiffler [Quivers with relations arising from clusters (A_n case), Trans. Amer. Math. Soc.358(3) (2006) 1347–1364], which we recover when p = m = 1, and the work of Baur–Marsh, [A geometric description of the m-cluster categories, Trans. Amer. Math. Soc.360(11) (2008) 5789–5803], treating the case p = 1, m > 1. Our approach also leads to a geometric model of the bounded derived category in type A.

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(x^n₀), x^n₁], x^n₂], …], x^n_k] = 0 for all x ∈ L, where n₀, n₁, n₂, …, n_k are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

Let G be a finite group and π_e(G) be the set of element orders of G. Assume that k ∈ π_e(G) and m_k(G) is the number of elements of order k in G. Set nse(G) ≔ {m_k(G) | k ∈ π_e(G)}, we call nse(G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L₂(2^a) by its order |L₂(2^a)| and the set nse(L₂(2^a)), where either 2^a - 1 or 2^a + 1 is a prime.

The isomorphism types of all minimal nonabelian subgroups (-subgroups) of -groups are described. This allows us to classify the nonabelian p-groups, p > 2, that have no p isomorphic -subgroups of minimal order. In particular, if a p-group G is neither abelian nor -group and all its -subgroups are pairwise non-isomorphic, then either G ≅ SD₁₆, the semidihedral group of order 16, or G is a metacyclic 2-group of order ≥ 2⁵. We also show that if a p-group G is neither abelian nor minimal nonabelian, then G is metacyclic if and only if all its -subgroups are metacyclic.

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

Zonotopal algebra studies pairs of dual algebraic structures that are associated with a linear matroid X and are connected to corresponding dual geometries. Both the geometry and the algebra encode in their statistics combinatorial properties of the matroid. We provide in this paper a general, unified, framework for the chapter in zonotopal algebra that is known as external. The approach is critically based on employing simultaneously the two dual algebraic constructs and invokes the underlying matroidal and geometric structures in an essential way.

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.

Main injective modules, which determine every left exact preradical, were introduced in a former work. In this paper, we consider those modules which determine every preradical and we call them main modules. We prove that a main module exists if and only if the lattice of preradicals R-pr is a set, and in this case we give a general construction. Some properties of main modules are proven. We also prove some characterizations of rings for which (a) every preradical is left exact, (b) every preradical is idempotent, (c) every preradical is a radical, (d) every preradical is a t-radical, (e) every preradical which is not the identity functor is prime. These characterizations relate to semisimple artinian rings, rings that are a direct product of a finite number of simple rings, left V-rings, simple rings, among others. In order to illustrate the theory introduced in this paper, several examples are provided.

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows:

(1) If gcd(p-1, |G|) = 1 and p² does not divide |x^G| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/O_p(G) is elementary abelian.

(2) Suppose that G is p-solvable. If p^p-1 does not divide |x^G| for any element x of prime power order, then the p-length of G is at most one.

(3) Suppose that G is p-solvable. If p^p-1 does not divide χ(1) for any χ ∈ Irr(G), then both the p-length and p′-length of G are at most 2.