Journal of Algebra and Its Applications

For a field 𝔽 and an integer d ≥ 1, we consider the universal associative 𝔽-algebra A generated by two sets of d + 1 mutually orthogonal idempotents. We display four bases for the 𝔽-vector space A that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in A looks with respect to our bases. Using our bases we obtain an infinite nested sequence of two-sided ideals for A. Using our bases we obtain an infinite exact sequence involving a certain 𝔽-linear map ∂: A → A. We obtain several results concerning the kernel of ∂; for instance this kernel is a subalgebra of A that is free of rank d.

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.

In 1996, Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki–Juriaans–Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a ℤ-order with hyperbolic unit group. In this paper, we complete this classification and give an easy proof that deals with all finite semigroups.

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.

Given a finite dimensional F-central simple algebra A = M_n(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

Recently the strings and the string number of self-maps were used in the computation of the algebraic entropy of specific abelian group endomorphisms. We introduce two special kinds of strings, and their relative string numbers. We show that a dichotomy holds for all these three string numbers; in fact, they admit only zero and infinity as values on abelian group endomorphisms.

Let M be a finitely generated module on a local ring R and a filtration of submodules of M such that d_o < d₁ < ⋯ < d_t = d, where d_i = dim M_i. This paper is concerned with a non-negative integer which is defined as the least degree of all polynomials in n₁, …, n_d bounding above the function

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

A cover for a group G is a collection of proper subgroups of G whose union is G. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Determining σ is an open problem for most non-solvable groups. We obtain σ(G) for some affine primitive groups.

The modules of the title are introduced as a proper generalization of quasi co-Hopfian modules, and they are characterized in several ways. The ring R is right nonsingular if and only if the densely co-Hopfian right R-modules and quasi co-Hopfian right R-modules coincide. Dense co-Hopficity is investigated for certain modules that have indecomposable decompositions complementing direct summands. For some classes of rings R, including rings with dense right socles, we determine when the injective envelope of an R-module is densely co-Hopfian.

Various authors classified the thick triangulated ⊗-subcategories of the category of compact objects for appropriate compactly generated tensor triangulated categories by using supports of objects. In this paper we introduce R-supports for ring objects, showing that these completely determine the thick triangulated ⊗-subcategories. R-supports give a general framework for the celebrated classification theorems by Benson–Carlson–Rickard–Friedlander–Pevtsova, Hopkins–Smith and Hopkins–Neeman–Thomason.

We deduce a recursive system of partial differential equations from the toroidal Lie algebra associated to a simple Lie algebra. Such a recursive system starts from the corresponding classical Toda field equations. For each truncated system we describe its Hamiltonian formalism using Gel'fand–Dikii's differential polynomial algebras, and some nontrivial local integrals are given.

Let f₀, f₁, f₂, f₃ be linearly independent nonzero homogeneous polynomials in the standard ℤ-graded ring R ≔ 𝕂[s, t, u] of the same degree d, and gcd(f₀, f₁, f₂, f₃) = 1. This defines a rational map ℙ² → ℙ³. The Rees algebra Rees(I) = R ⊕ I ⊕ I² ⊕ ⋯ of the ideal I = 〈f₀, f₁, f₂, f₃〉 is the graded R-algebra which can be described as the image of the R-algebra homomorphism h: R[x, y, z, w ] → Rees(I). This paper discusses one result concerning the structure of the kernel of the map h when I is a saturated local complete intersection ideal with V(I) ≠ ∅ and μ-basis of degrees (1,1,d - 2).