Algebra Colloquium

Verma modules over the W-algebra W(2,2) were considered by Zhang and Dong, while the Harish-Chandra modules and irreducible weight modules over this algebra were classified by Liu and Zhu etc. In the present paper, we investigate the Lie bialgebra structures on the centerless W-algebra, which are shown to be triangular coboundary.

In this paper, we give a characterization and a complete classification of minimal-class-3 p-groups, that is, the p-groups of class 3 all of whose proper sections have class at most 2.

The structure of locally graded groups with finitely many normalizers of non-polycyclic subgroups is investigated. In particular, it is proved that such groups either are polycyclic or have Černikov commutator subgroups.

In this paper, we show that if J is a complex Jordan Banach algebra in which inner ideals are countably generated, then J is finite dimensional. We also study the real case.

We consider the problem of classifying those groups whose maximal cyclic subgroups are maximal. We give a complete classification of those groups with this property and which are either soluble or residually finite.

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties , we prove that S and ΨS simultaneously satisfy or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice of varieties of completely regular semigroups regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on and evaluate it on a small sublattice of .

In this paper, we study the self-complementary Cayley graphs and count the number of isomorphic classes of undirected self-complementary circulants of order n, where n is square free.

In this paper, we complete the classification of Z-graded modules of the intermediate series over a q-analog Virasoro-like algebra L. We first construct four classes of irreducible Z-graded L-modules of the intermediate series. Then we prove that any Z-graded L-module of the intermediate series must be one of the modules constructed by us, or a direct sum of some trivial L-modules.

An endomorphism α of a ring R is called semicommutative if ab = 0 implies aR α(b) = 0 for a, b ∈ R, and R is called an α-semicommutative ring if there exists a semicommutative endomorphism α of R. The notion of an α-semicommutative ring is an extension of semicommutative rings as well as a generalization of α-rigid rings. We investigate its related properties and the relationship between α-semicommutative rings and extended Armendariz rings.

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and 𝔤 = Lie(G). In this paper, we study representations of the reductive Lie algebra 𝔤 with p-character χ of standard Levi form associated with an index subset I of simple roots. With aid of the support variety theory, we prove that a U_χ(𝔤)-module is projective if and only if it is a strong "tilting" module, i.e., admitting both - and -filtrations. Then by an analogy of the arguments in [2] for G₁T-modules, we construct so-called Andersen–Kaneda filtrations associated with each projective 𝔤-module of p-character χ, and finally obtain sum formulas from those filtrations.

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.

Let R be a prime ring with a nontrivial idempotent. In this paper, we prove that if g is an additive map of R into itself such that xg(y)z = 0 for all x, y, z ∈ R with xy = yz = 0, then g is a generalized derivation. As an application of this result, we show that every local generalized derivation in a prime ring with a nontrivial idempotent is a generalized derivation.

We define and investigate (structural) row-finite matrices, lower triangular matrices, and power series in near-rings by using inverse limits of some special classes of matrices. Our approach is different from that described in [8]. We show that polynomials can be embedded in power series and power series can be embedded in lower triangular matrices as in rings. We extend the concept of order of a polynomial (or a power series) from rings to near-rings. A natural topology is defined on lower triangular matrices by generalizing the concept of order of power series. We also obtain some results concerning prime and completely prime ideals in lower triangular matrix near-rings.

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x₁,…,x_n) a non-central multilinear polynomial over K, d and δ derivations of R, a and b fixed elements of R. Denote by f(R) the set of all evaluations of the polynomial f(x₁,…,x_n) in R. If a[d(u),u] + [δ (u),u]b = 0 for any u ∈ f(R), we prove that one of the following holds: (i) d = δ = 0; (ii) d = 0 and b = 0; (iii) δ = 0 and a = 0; (iv) a, b ∈ Z(R) and ad + bδ = 0. We also examine some consequences of this result related to generalized derivations and we prove that if d is a derivation of R and g a generalized derivation of R such that g([d(u),u]) = 0 for any u ∈ f(R), then either g = 0 or d = 0.

An element a in a ring R is called left morphic if R/Ra ≅ ℓ_R(a), where ℓ_R(a) denotes the left annihilator of a in R. A ring R is said to be left morphic if every element is left morphic. In this paper, it is shown that if I is an ideal of a unit regular ring R, then for each positive integer n,

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X₁, X₂, X₃, X₄ in solution X = X₁ + X₂i + X₃j + X₄k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)^- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A₁XB₁ = C₁ and A₂XB₂ = C₂ to have a common real solution.