Journal of Algebra and Its Applications

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group C_n. In this paper, we investigate the minimum value of ψ on the set of groups of the same order.

In this note, derivations on Morita rings (generalized matrix rings) are introduced and generalized derivations on rings are proved to be Morita invariant without involvement of any homology theory.

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL(2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

Is it possible for a particular Costas array to be generated by two different constructions of the Golomb and Welch families? Experimental data suggests that this does not happen (except for trivially small orders), and a (partial) proof of this fact is offered herein through a case-by-case study of all possible pairs of constructions that can potentially produce Costas arrays of equal order.

In continuation of our recent work about the quasitriangular structures for the twisted tensor biproduct, we give the necessary and sufficient conditions for Brzeziński crossed coproduct coalgebra, including the twisted tensor coproduct introduced by Caenepeel, Ion, Militaru and Zhu and crossed coproduct as constructed by Lin, equipped with the usual tensor product algebra structure to be a Hopf algebra. Furthermore, the necessary and sufficient conditions for Brzeziński crossed coproduct to be a quasitriangular Hopf algebra are obtained.

In the present work, binary-Lie, assocyclic, and binary (-1,1) algebras are studied. We prove that, for every assocyclic algebra A, the algebra A^- is binary-Lie. We find a simple non-Malcev binary-Lie superalgebra that cannot be embedded in A^-s for an assocyclic superalgebra A. We use the Grassmann envelope of to prove the similar result for algebras. This solve negatively a problem by Filippov (see [1, Problem 2.108]). Finally, we prove that the superalgebra is isomorphic to the commutator superalgebra for a simple binary (-1,1) superalgebra .

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R_𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R_𝔪. As a corollary, we obtain the computability of the category of finitely presented R_𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A₅, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and M be an arbitrary R-module. In this paper, among other things, we show that if, for a non-negative integer t, the R-module is weakly Laskerian and is 𝔞-weakly cofinite for all i < t, then, for any weakly Laskerian submodule U of , the R-module is weakly Laskerian. As a consequence the set of associated primes of is finite.

Let G be a finite group. The main result of this paper is as follows: if G is a finite group such that Γ(G) = Γ(M), where M is G₂(q) (q = 3²ⁿ⁺¹) or ²B₂(q) (q = 2²ⁿ⁺¹ > 2), then G is quasirecognizable by prime graph. Hence we generalize some known results of G₂(q) and ²B₂(q).

A set of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are:

(1) Every irreducible semitransitive Jordan algebra is actually transitive.

(2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R_[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

The main result of this paper gives several characterizations for Abelian groups A to have the property that the class of A-adstatic modules is closed with respect to submodules of bounded index. Especially, a group has this property if and only if it is flat and faithful with respect to torsion when viewed as a left module over its endomorphism ring.

We shall define the associative algebra structure closest to a given algebra structure. When we face a new multiplication, being caused by noise and so on, it must be useful to compute with the closest associative multiplication to such a perturbed one. In this paper we shall give a procedure to find the closest associative structure and demonstrate our strategy for the 2-dimensional algebras over the field of real numbers. Then we trace the points of the sets of structure constants of the deformed algebras.