Algebra Colloquium

We deduce a ℤ-basis for the cluster algebra of type . It extends the results obtained by Caldero and Keller, as well as Sherman and Zelevinsky. Moreover, we prove that coefficients of Laurent expansions in the generalized cluster variables for are positive integers.

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F: A → M is said to be range-inclusive (respectively, skew range-inclusive) if [A,F(x)] ⊆ [x,M] for every x ∈ A (respectively, A ◦ F(x) ⊆ x ◦ M for every x ∈ A). Recently, Brešar investigated range-inclusive maps from a unital ring with wide idempotents into a unital A-bimodule. In this paper, our main purpose is to investigate skew range-inclusive maps from a unital ring with wide idempotents into a unital 2-torsion free A-bimodule.

We study a class of subalgebras of the Lie algebra of the derivations on a torus. We construct a functor F^α from linear Lie algebra modules to these subalgebra modules. We then characterize the structure of the image modules of corresponding linear Lie algebra modules under this functor.

It is well known that the m × m upper triangular matrix ring over any ring is not ZI_n (and so not ZC_n) for m ≥ 2. In this paper, we find some ZC_n subrings and ZI_n subrings of the upper triangular matrix ring over a reduced ring.

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.

The concepts of soft rings and idealistic soft rings are introduced. Three basic isomorphism theorems for soft rings are established, and consequently, some properties of soft rings and idealistic soft rings are given.

Let G be a finite group. A subgroup H of G is called a 2-maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. In this paper, we discuss the influence of π-quasinormality of 2-maximal subgroups of Sylow subgroups on the structure of a finite group, and obtain some sufficient conditions under which the finite group is p-nilpotent, supersolvable, or possesses an ordered Sylow tower.

Let be the toric ideal of a homogeneous normal configuration . We prove that is generated by circuits if and only if each unbalanced circuit of has a connector which is a linear combination of circuits with a square-free term. In particular, if each circuit of with non-square-free terms is balanced, then is generated by circuits. As a consequence, we prove that the toric ideal of a normal edge subring of a multigraph is generated by circuits with a square-free term.

Let Λ be a finite-dimensional superalgebra over a field K. A characterization of an elementary superalgebra Λ is given by a quiver and a weight function. It is shown that Λ is elementary if and only if its Hochschild extension is elementary. Furthermore, if Λ is elementary of finite global dimension and {e₁, …, e_n} is a complete set of gr-primitive orthogonal idempotents of Λ, then the following equalities hold:

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module is weakly Laskerian for all i < r. Then we prove that is also weakly Laskerian and so is finite. Moreover, we show that if s is a non-negative integer such that is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of is weakly Laskerian?

A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that AB^x=B^xA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.

Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

In this paper, we introduce the notion of a modular pull-back exact category, study modular lattices and on the skeletally small modular pull-back exact category (𝒞,ℰ) and show that there is an isomorphism between these two lattices. We also study short exact sequences of lattices induced by ℰ-exact sequences.

Let Θ be a collection of formations. A formation 𝔉 ∈ Θ is called an ℌ_Θ-critical formation or a minimal non-ℌ Θ-formation if 𝔉 ⊈ ℌ but the class ℌ of groups contains each proper Θ-subformation of 𝔉. In this paper we study ℌ_Θ-critical formations, where Θ is the class of all τ-closed ω-local formations.

Let 𝔽 be a field of characteristic 0, G an additive subgroup of 𝔽, s ∈ 𝔽 such that s ∉ G and 2s ∈ G. A class of infinite-dimensional Lie algebras called generalized Schrödinger-Virasoro algebras was defined by Tan and Zhang, which is a natural generalization of Schrödinger-Virasoro algebras. In this paper, central extensions and derivations of are determined.

In this paper, we realize Verma modules and the vector-coherent-state (VCS) representations of the orthosymplectic Lie superalgebra osp(2|2n) as differential operators with vector coefficients. We also characterize simple sub-representations of VCS representations as kernels of some finite number of differential operators. The singular vectors of the atypical representation of osp(2|2n) are explicitly given.

A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-r finite-dimensional simple Lie algebra is exactly 2^r. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite-dimensional simple Lie algebras, which is a refinement of Peterson's theorem capturing more Lie algebra invariants.