Journal of Algebra and Its Applications

Some properties of (left) k-ideals and r-ideals of a semiring are considered by the help of the congruence class semiring. It is proved that a proper k-ideal of a semiring with an identity is prime if it is a maximal left k-ideal. An equivalent condition for a proper r-ideal of a semiring being a maximal (left) r-ideal is established. It is shown that (left) r-ideals and (left) k-ideals coincide for an additively idempotent semiring, though the former is a special kind of the latter in general. It is proved that a proper k-ideal of an incline with an identity is a maximal k-ideal if and only if the corresponding congruence class semiring is the Boolean semiring.

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by 𝔍_R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍_R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍_R) denote the clique number of 𝔍_R. It was conjectured that if is a commutative finite ring and (R_i, 𝔪_i) is a local ring, for i = 1, …, n, then , where F_i = R_i/𝔪_i, for i = 1, …, n. In this paper, we prove that if R is a commutative ring (not necessarily finite) and R is not a field, then ω(𝔍_R) = max_{𝔪∈Max(R)} |𝔪| and using this we show that the aforementioned conjecture holds.

In this paper, we give some necessary and sufficient conditions of a module to be an almost Cohen–Macaulay module by using a characterization of the associated primes set of the first non-vanishing local cohomology module. Also, we give a class of examples of almost Cohen–Macaulay modules.

We study a class of generalized Kac–Moody algebras which are fixed point subalgebras of the covering Kac–Moody algebras of some generalized intersection matrix (GIM) algebras of Slodowy.

We show that a non-hyperbolic quadratic pair on a central simple algebra Brauer equivalent to a quaternion algebra stays non-hyperbolic over some splitting field of the quaternion algebra. This extends a result previously only known for fields of characteristic different from two. Our presentation is free from restrictions on the characteristic of the base field.

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism.

We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture.

In the group generated by automorphisms and anti-automorphisms of A₁, all the involutions belong to one conjugacy class, hence:

• Every involutive endomorphism from (A₁, γ) to (A₁, δ) is an automorphism (γ and δ are two involutions on A₁).

• Given an endomorphism f of A₁ (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A₁), then f is an automorphism.

In this paper we prove that standardly based algebras are invariant under Morita equivalences. As an application, we prove 0-Hecke algebras and 0-Schur algebras are standardly based algebras. From this point of view, we give a new way to construct the simple modules of 0-Hecke algebras, and prove that the dimension of the center of a symmetric 0-Hecke algebra is not less than the number of its simple modules.

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. Following Cuong and the first author [N. T. Cuong and L. T. Nhan, J. Algebra267 (2003) 156–177], M is called pseudo Cohen–Macaulay if for a system of parameters of M, where . In this paper, first we improve some known results on pseudo Cohen–Macaulay modules. Then we study the localizations of pseudo Cohen–Macaulay modules in order to introduce the pseudo Cohen–Macaulayness for the non-local case. Finally, we give characterizations for the formal power series ring and the polynomial ring being pseudo Cohen–Macaulay.

If 𝔛 is a class of groups, a group G is minimal non-𝔛 if it is not an 𝔛-group, but all its proper subgroups belong to 𝔛. The aim of this paper is to prove that for an infinite locally graded group, the property of being minimal non-hypercentral and that of being minimal non-hypercyclic are equivalent. Moreover, the main properties of infinite minimal non-hypercentral groups are described. In the last section, we study groups of infinite rank in which all proper subgroups of infinite rank satisfy a generalized supersolubility condition.

In 1981, S. V. Pchelintsev introduced the notion of topological rank for Spechtian varieties of algebras as a certain tool for studying the structure of non-nilpotent subvarieties in a given variety. We provide a variety of right alternative algebras of arbitrary given finite topological rank. Namely, we prove that the topological rank of the variety of right alternative metabelian (solvable of index two) algebras that are Lie-nilpotent of step not more than s over a field of characteristic distinct from two and three is equal to s.

In this paper, we introduce the concepts of semi-Baer, semi-quasi Baer, semi-p.q. Baer and semi-p.p. modules as a generalization of Baer, quasi Baer, p.q. Baer and p.p. modules respectively. To define these concepts, we introduce concepts of multiplicative order of an element and a multiplicatively finite element in rings. Further, we characterize these concepts in modules over reduced rings. Also, it is proved that semi-Baer and semi-quasi Baer properties are preserved by polynomial extensions and power series extensions of modules. It is proved that for a ring R and a monoid G, if the semi group ring RG is semi-Baer (semi-quasi Baer) then so is R.

Let G be a finite group. We define the subgroup N^𝒩_p(G) to be the intersection of the normalizers of the p-nilpotent residuals of all subgroups of G for a prime p. We first investigate the properties of N^𝒩_p(G), and then study the relationship between ⋂_p‖G∣N^𝒩_p(G) and N^𝒩(G), where N^𝒩(G) is the intersection of the normalizers of the nilpotent residuals of all subgroups of G.

For a special class of generalized Weyl algebras (GWAs), we prove a Duflo theorem stating that the annihilator of any simple module is in fact the annihilator of a simple highest weight module.

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are PSL₃(q), q ≠ 4, and PSU₃(q²), q² ≠ 9, 25. If n = 5 they are PSL₅(q), for all q, and PSU₅(q²), q² ≥ 9. Also, the soluble group PSU₃(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups.

We study an associative algebra A over an arbitrary field that is a sum of two subalgebras A₁ and A₂ (i.e. A = A₁ + A₂). Additionally we assume that A_i has an ideal of finite codimension in A_i which satisfies a polynomial identity f_i = 0 for i = 1, 2. Suppose that all rings R = R₁ + R₂, which are sums of subrings R₁ and R₂, are PI rings when R_i satisfies the polynomial identity f_i = 0 for i = 1, 2. We prove that A is a PI algebra.

Let R be an associative ring with identity and Spec^s(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Spec^s(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Spec^s(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Spec^s(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Spec^s(M) is connected.