Algebra Colloquium

Let G be an algebraic supergroup over a field k, whose pure-even group scheme G_ev is a reduced algebraic group scheme, i.e., is reduced. In this paper, we prove that Lie(G) can be identified with the Lie superalgebra of admissible left-invariant derivations of k[G].

By a combined use of analytical, algebraic and computational tools, we derive a description of the algebra of modular forms with respect to a certain congruence subgroup of SL₂(ℤ) × SL₂(ℤ) of level 3.

We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.

In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M-p-injective module which is not M-pseudo injective is given.

In this paper, we study the t-dimension of a pullback and show that for any pair of integers (n,m) with 1 ≤ n ≤ m ≤ 2n-1, there is a domain R such that t-dim(R)=n and t-dim(R[X])=m.

Among other results on subtractive semimodules and semirings, we present various (homological) characterizations of subtractive semisimple semirings. Also, we give complete descriptions of finite subtractive semisimple as well as additively regular (in particular, additively idempotent) subtractive semisimple semirings.

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 𝔇_(n)(G) and 𝔇_[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 𝔇_(n)(G) =𝔇_[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG.

Recently, in [25], Lie bialgebra structures on Lie algebras of generalized Weyl type were considered, which are shown to be triangular coboundary. In this paper, we quantize these algebras with their Lie bialgebra structures.

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type B_n and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of S_n which correspond to partitions with all parts odd. For the classes of S_n which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

The idempotent quiver of a nearring R is a directed graph formed using primitive idempotents of R corresponding to the isomorphism classes of the type 2 modules of R. In this paper, we construct such quivers when R satisfies the descending chain condition on right R-subgroups and J₂(R) is nilpotent, consider their computation, and study their relationship with the structure of R and its modules.

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety , all mapping extensions by φ of algebras from form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

A ring R is called right McCoy if whenever polynomials f(x), g(x) ∈ R[x]∖{0} satisfy f(x)g(x)=0, there exists a nonzero r ∈ R such that f(x)r=0. We continue in this paper the study of right McCoy rings by Nielsen [8]. We first consider properties and basic extensions of right McCoy rings, providing many examples in the process. Next, we show that if R is a right McCoy ring, then R[x] and R[x]/(xⁿ) are right McCoy rings, where (xⁿ) is the ideal generated by xⁿ and n is a positive integer; and that for a right Ore ring R with Q its classical right quotient ring, R is right McCoy if and only if Q is right McCoy.

The maximal subalgebras and their intersection of a Lie algebra or a Lie superalgebra were studied by Racine, Scheiderer, Elduque, Melikyan, et al. The purpose of the present paper is to continue the investigation in order to obtain deeper structure theorems for Lie superalgebras. We develop the Frattini theory for Lie superalgebras, generalize Barnes's results to Lie superalgebras, and obtain some necessary and sufficient conditions for solvable Lie superalgebras and nilpotent Lie superalgebras. Moreover, some necessary and sufficient conditions for ϕ-free Lie superalgebras and elementary Lie superalgebras are given.

The first part of this paper is concerned with the Artinianness of certain local cohomology modules when M is a Matlis reflexive module over a commutative Noetherian complete local ring R and 𝔞 is an ideal of R. Also, we characterize the set of attached prime ideals of , where n is the dimension of M. The second part is concerned with the vanishing of local cohomology and generalized local cohomology modules. In fact, when R is an arbitrary commutative Noetherian ring, M is an R-module and 𝔞 is an ideal of R, we obtain some lower and upper bounds for the cohomological dimension of M with respect to 𝔞.

A Hamilton cycle in a graph is a cycle going through all vertices of the graph, and a graph is said to be a Hamilton graph if it has a Hamilton cycle. In this article, two sufficient conditions for vertex-transitive Hamilton graphs of prime-power order are given. Using these conditions, two infinite families of Hamilton graphs of order a 2-power are constructed.

A group G=AB is said to be totally factorized by its subgroups A and B if XY=YX for all subgroups X of A and Y of B. It is known that any finite group totally factorized by supersoluble subgroups is supersoluble, and that a finite group totally factorized by nilpotent subgroups is abelian-by-nilpotent. This latter result is extended here to certain classes of infinite groups.