Many variations of the notions of clean and strongly clean have been studied by a variety of authors. We develop a general theory, based on idempotents and direct sum decompositions, that unifies several of these existing concepts. As a specific case, we also investigate a new class of clean rings.

Let sigma = {sigma(i)vertical bar i is an element of I} be some partition of the set P of all primes, that is, P = U-i is an element of I sigma(i) and sigma(i) boolean AND sigma(j) = empty set for all i not equal j. Let G be a finite group. We say that G is: sigma-primary if G is a sigma(i)-group for some i is an element of I; sigma-soluble if every chief factor of G is sigma-primary. We say that a set H = {H-1, ... , H-t} of Hall subgroups of G, where H-i is sigma-primary (i = 1, ... ,t), is a complete Hall set of type sigma of G if (vertical bar H-i vertical bar, vertical bar H-j vertical bar) = 1 for all i not equal j and pi(G) = pi(H-1) boolean OR ... boolean OR pi(H-t). We say that a subgroup A of G is: sigma-subnormal in G if there is a subgroup chain A = A(0) <= A(1) <= ... <= A(n) = G such that either A(i-1) is normal in A(i) or A(i)/(A(i-1))A(i) is sigma-primary for all i = 1, ... ,t; sigma-permutable in G if G has a complete Hall set H of type sigma such that A is H-G-permutable in G, that is, AH(x) = H(x)A for all x is an element of G and all H is an element of H. We study the relationship between the sigma-subnormal and sigma-permutable subgroups of G. In particular, we prove that every sigma-permutable subgroup of G is sigma-subnormal, and we classify finite sigma-soluble groups in which every sigma-subnormal subgroup is sigma-permutable. (C) 2015 Elsevier Inc. All rights reserved.

Let be some partition of the set of all primes, that is, and for all . Let be a finite group. We say that is: if is a -group for some ; if every chief factor of is -primary. We say that a set of Hall subgroups of , where is -primary ( ), is a of if for all and . We say that a subgroup of is: in if there is a subgroup chain such that either is normal in or is -primary for all ; in if has a complete Hall set of type such that is -permutable in , that is, for all and all . We study the relationship between the -subnormal and -permutable subgroups of . In particular, we prove that every -permutable subgroup of is -subnormal, and we classify finite -soluble groups in which every -subnormal subgroup is -permutable.

Let be a finite group, a subgroup of and the subgroup of generated by all those subgroups of which are -permutable in . Then we say that is weakly -permutable in if has a subnormal subgroup such that and . We fix in every non-cyclic Sylow subgroup of a subgroup satisfying and study the structure of under the assumption that all subgroups with are weakly -permutable in .

Let be a commutative ring with its ideal of nilpotent elements, its set of zero-divisors, and its set of regular elements. In this paper, we introduce and investigate the of , denoted by . It is the (undirected) graph with all elements of as vertices, and for distinct , the vertices and are adjacent if and only if . We also study the three (induced) subgraphs , , and of , with vertices , , and , respectively.

Let sigma = {sigma(i) broken vertical bar i is an element of I} be some partition of the set of all primes P and G a finite group. G is said to be sigma-soluble if every chief factor H/K of G is a sigma i-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall sigma-set of G if every member not equal 1 of H is a Hall sigma(i)-subgroup of G for some sigma(i) is an element of sigma and H contains exactly one Hall sigma(i)-subgroup of G for every i is an element of I such that sigma(i) boolean AND pi(G) not equal (sic). A subgroup A of G is said to be sigma-permutable in G if G has a complete Hall sigma-set H such that AH(x) = X-x for all x is an element of G and all H is an element of H. We obtain characterizations of finite sigma-soluble groups G in which sigma-permutability is a transitive relation in G. (C) 2017 Elsevier Inc. All rights reserved.

Given a finite-dimensional reductive Lie algebra equipped with a nondegenerate, invariant, symmetric bilinear form , let denote the universal affine vertex algebra associated to and at level . Let be a vertex (super)algebra admitting a homomorphism . Under some technical conditions on , we characterize the coset for generic values of . We establish the strong finite generation of this coset in full generality in the following cases: , , and . Here and are finite-dimensional Lie (super)algebras containing , equipped with nondegenerate, invariant, (super)symmetric bilinear forms and which extend , is fixed, and is a free field algebra admitting a homomorphism . Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple superconformal algebra with for all positive integers .

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the noncommutative realm, -algebras with a dualizing complex.

For any characteristic zero coefficient field, an irreducible representation of a finite -group can be assigned a Roquette -group, called the genotype. This has already been done by Bouc and Kronstein in the special cases and . A genetic invariant of an irrep is invariant under group isomorphism, change of coefficient field, Galois conjugation, and under suitable inductions from subquotients. It turns out that the genetic invariants are precisely the invariants of the genotype. We shall examine relationships between some genetic invariants and the genotype. As an application, we shall count Galois conjugacy classes of certain kinds of irreps.

For any prime p, all constacyclic codes of length p(s) over the ring R = F-pm + uF(p)m are considered. The units of the ring R are of the forms gamma and alpha + u beta, where alpha, beta, and gamma are nonzero elements of F-pm. which provides pm(pm 1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length p(s) over the finite field F-pm are obtained: they are used as a tool to establish the structure and Hamming distances of all (alpha + u beta)-constacyclic codes of length p(s) over R. We then classify all cyclic codes of length p(s) over R and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and gamma-constacyclic codes of length 195 over R is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to gamma-constacyclic codes of length p(s) over R. (C) 2010 Elsevier Inc. All rights reserved.

I describe the current state of the development version of the package, which allows to transform the theories of Coxeter groups, reductive algebraic groups, complex reflection groups, Hecke algebras, braid monoids, etc. … into actual computations. Examples are given, showing the code to check some results of Lusztig.

The study of singularities of algebraic or analytic spaces over the field of complex numbers is a traditional subject, but, in contrast, the parallel theory over arbitrary algebraically closed fields is still poor and there are lots of interesting questions to be answered. Our main concern here is the study of the Milnor number of an isolated hypersurface singularity which is defined as the codimension of the ideal generated by the partial derivatives of a power series that represents locally the hypersurface. This is an important topological invariant of the singularity over the complex numbers, but its behavior changes dramatically when the base field has positive characteristic, in which case it may be infinite and depends upon the local equation of the hypersurface, not being an intrinsic invariant. In this paper we will study the variation of the Milnor number in terms of the local equations, showing that its minimum value has an intrinsic meaning and give necessary and sufficient conditions for its invariance. We also relate the smoothness of the generic fiber of an isolated hypersurface singularity deformation with the finiteness of this number, connecting it to a Bertini type result. At the end, we will show how, in arbitrary characteristic, one defines the sequence of Milnor numbers of sections of a hypersurface singularity by general linear spaces of increasing codimension.

A criterion for elements of free Zinbiel algebras to be Lie or Jordan is established. This criterion is used in studying speciality problems of Tortkara algebras. We construct a base of free special Tortkara algebras. Furthermore, we prove analogues of classical Cohn's and Shirshov's theorems for Tortkara algebras.

We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove the duality property, weak associativity, a Jacobi identity, a generalized commutator formula, generalized weak commutativity, and convergence and commutativity for products of more than two operators involving twist vertex operators. These properties of twist vertex operators play an important role in the author's recent general, direct and explicit construction of (lower-bounded generalized) twisted modules.

We extend the Dong-Mason theorem on irreducibility of modules for orbifold vertex algebras (cf. ) to the category of weak modules. Let be a vertex operator algebra, an automorphism of order . Let be an irreducible weak –module such that are inequivalent irreducible modules. We prove that is an irreducible weak –module. This result can be applied on irreducible modules of certain Lie algebra such that are Whittaker modules having different Whittaker functions. We present certain applications in the cases of the Heisenberg and Weyl vertex operator algebras.

Let and be finite groups having coprime orders and suppose that acts on via automorphisms. We give some solvability criteria for according to the number of orbits that appear by the action of the fixed point subgroup on the set of maximal -invariant subgroups of , and likewise, on the set of non-nilpotent maximal -invariant subgroups. We also obtain some characterizations and further structure properties of these groups. In the course of our study we prove an independent result concerning maximal factorizations of classical simple groups.

Let be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field . In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category . Let be the Majid's transmuted braided group of , we show that is cosemisimple. As a coalgebra, let be the sum of minimal -adjoint-stable subcoalgebras. For each , we choose a minimal left coideal of , and we can define the -adjoint-stable algebra of . Using Ostrik's theorem on characterizing module categories over monoidal categories, we prove that is irreducible if and only if there exists an and an irreducible right -module , such that . Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If is an algebraically closed field of characteristic 0, we stress that the -adjoint-stable algebra is an algebra over which the dimension of each irreducible right module divides its dimension.

Let be a commutative cancellative monoid, and let be an integral domain. The question of whether the monoid ring is atomic provided that both and are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for : he constructed an atomic integral domain such that the polynomial ring is not atomic. However, the question of whether a monoid algebra over a field is atomic provided that is atomic has been open since then. Here we offer a negative answer to this question. First, we exhibit for any infinite cardinal a torsion-free atomic monoid of rank satisfying that the monoid domain is not atomic for any integral domain . Then for every and for each field of finite characteristic we find a torsion-free atomic monoid of rank such that is not atomic. Finally, we construct a torsion-free atomic monoid of rank 1 such that is not atomic.

We construct two supercharacter theories (in the sense of P. Diaconis and I.M. Isaacs) for the parabolic subgroups in orthogonal and symplectic groups. For each supercharacter theory, we obtain a supercharacter analog of the A.A. Kirillov formula for irreducible characters of finite unipotent groups.