A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra R_G. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that R_G is closely related to the symmetric algebra over F_p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R_G in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of R_G is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.     

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J (R) (or CGJ (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.     

Zero-divisor graphs of amalgamated duplication of a ring along an ideal

Let R be a commutative ring with identity and let I be an ideal of R. Let R?I be the subring of R×R consisting of the elements (r,r+i) for r∈R and i∈I. We study the diameter and girth of the zero-divisor graph of the ring R?I.     

Generalized cover ideals and the persistence property

Let I   be a square-free monomial ideal in R=k[_x1,…,_xn]$R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes Ass(I^s)$\mathrm{Ass}(I^s)$ for all integers s?1$s?1$. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G   is a tree, we explicitly determine Ass(I^s)$\mathrm{Ass}(I^s)$ for all s?1$s?1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.     

Embedded associated primes of powers of square-free monomial ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than β₁, where β₁ is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If, in addition, I fails to have the packing property, then embedded primes of I^t do occur when t=β₁+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.     

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.     

TW-domains and Strong Mori domains

In this paper we are mainly concerned with TW-domains, i.e., domains in which the w- and t-operations coincide. Precisely, we investigate possible connections with related well-known classes. We characterize the TW-property in terms of divisoriality for Mori domains and Noetherian domains. Specifically, we prove that a Mori domain R is a TW-domain if and only if R_M is a divisorial domain for each t-maximal ideal M of R. It turns out that a Mori domain which is a TW-domain is a Strong Mori domain. The last section examines the transfer of the “TW-domain” and “Strong Mori” properties to pullbacks, in order to provide some original examples.     

Constructing Witt-Burnside rings

The Witt-Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt-Burnside ring of G over the monoid ring Z[M], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A_× denotes the set A as a monoid under multiplication, then the Witt-Burnside ring of G over Z[A_×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt-Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher [Adv. in Math. 70 (1988) 87-132] and Graham [Adv. in Math. 99 (1993) 248-263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings.     

Mod 2 indecomposable orthogonal invariants

Over an algebraically closed base field k of characteristic 2, the ring R^G of invariants is studied, G being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring R of the m-fold direct sum kⁿ⊕?⊕kⁿ of the standard vector representation. It is proved for O(n) (n?2) and for SO(n) (n?3) that there exist m-linear invariants with m arbitrarily large that are indecomposable (i.e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of m. Indecomposability of corresponding invariants over immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps endowed with the Whitney (graph) topology and by Cc(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l₂-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X,G) is an (R^∞×l₂)-manifold, and moreover the pair (C(X,G),Cc(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l₂.     

New upper bounds for Estrada index of bipartite graphs

Suppose G is a graph and λ₁,λ₂,…,λ_n are the eigenvalues of G. The Estrada index EE(G) of G is defined as the sum of e^λ_i, 1≤i≤n. In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a (4,6)-fullerene to improve our bound given in an earlier paper.     

A description of Baer-Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a₁,a₂,a₃∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}^′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).     

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

The inertia set of the join of graphs

Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[a_i,j] with a_i,j≠0, i≠j if and only if ij is an edge of G. Denote by I^↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G₁ and G₂, then I^↗(G)?{(1,1)}=I^↗(G₁∨K₁)∩I^↗(G₂∨K₁)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I^↗(G). We use these results to determine I^↗(G) for every complete multipartite graph G.     

Blow-up rings and rational singularities

Let A be a normal local ring which is essentially finite type over a field of characteristic zero. Let I⊂A be an ideal such that the Rees algebra R _A (I) is Cohen–Macaulay and normal. In this paper we address the question: “When does R _A (I) have rational singularities?” In particular, we study the connection between rational singularities of R _A (I) and the adjoint ideals of the powers I ⁿ (n∋ℕ). Received: 25 May 1998 / Revised version: 20 August 1998