Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

On Factorization in Modules

In two articles, Anderson and Valdes-Leon generalized the theory of factorization in integral domains to commutative rings with zero divisors and to modules. Here we investigate some factorization properties in modules and state a result that relates factorization properties of an R-module, M, to the factorization properties of M as an (R/Ann(M))-module. Furthermore, we will investigate when a polynomial module, M[x], has the bounded factorization property, assuming that M has this property.     

Group-Like Algebras and Their Representations

The concept of “group-like algebras” was defined by the author as a special class of bF algebras. They generalize scheme rings (Bose–Mesner algebras) of noncommutative association schemes. We develop the representation theory for group-like algebras and symmetric bF algebras. We also study group-like algebras of association-scheme type with dimension 2 and 3.     

On the Complement of the Dense Orbit for a Quiver of Type 𝔸

Let 𝔸 _t be the directed quiver of type 𝔸 with t vertices. For each dimension vector d, there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver 𝔸 _t and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit, we determine the irreducible components and their codimension. Finally, we consider several particular examples.     

Various Forms of Generating Subsemigroups in Algebraic Monoids

Let M be an irreducible affine algebraic monoid over an algebraically closed field, G its unit group, and E(M) the set of idempotents of M. We study various forms of subsemigroup generating in affine algebraic monoids and relevant generating problems with kernel data. We determine the structure of minimal irreducible algebraic submonoids containing the kernel, in particular, of M = G ∪ ker(M). We also prove that M with a dense unit group is regular if and only if M = ? E(M), G ? and ? E(M) ? is regular.     

Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

The famous irreducibility criteria of Schönemann–Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann–Eisenstein–Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.     

Representations of Generalized Almost-Jordan Algebras

This paper deals with the variety of commutative algebras satisfying the identity β{(yx ²)x ? ((yx)x)x} + γ{yx ³ ? ((yx)x)x} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel, and Piacentini-Cattaneo [6 Carini , L. , Hentzel , I. R. , Piacentini-Cattaneo , J. M. ( 1988 ). Degree four identities not implies by commutativity . Comm. in Algebra 16 ( 2 ): 339 – 357 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field is different from 2, 3.     

On the Character Degrees of Suzuki p-Groups of Type A and C

In this note, we construct the irreducible characters of Suzuki p-groups of types A _p (m, θ) and C _p (m, θ, ?).     

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(M_n), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(M_n) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(M_n).