Arbitrary Commutative Ring Research Articles

Commutative Rings Whose Zero-Divisor Graphs Have Positive Genus

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R U = A U then the conclusion is that A is integral over R.

On constacyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring, γ a generator of the maximal ideal and R× the multiplicative group of units of R. For any w∈R×, the structural properties and dual codes of (1+wγ)-constacyclic codes of arbitrary length over R are given. As corollaries, self-dual constacyclic codes over the finite chain ring F2m+uF2m (u2=0) and the Galois ring GR(2s,m) are provided.

On the center of fusion categories

Muger proved in 2003 that the center of a spherical fusion category C of nonzero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of C. We generalize this theorem to a pivotal fusion category C over an arbitrary commutative ring k, without any condition on the dimension of the category. (In this generalized setting, modularity is understood as 2-modularity in the sense of Lyubashenko.) Our proof is based on an explicit description of the Hopf algebra structure of the coend of the center of C. Moreover we show that the dimension of C is invertible in k if and only if any object of the center of C is a retract of a “free” half-braiding. As a consequence, if k is a field, then the center of C is semisimple (as an abelian category) if and only if the dimension of C is nonzero. If in addition k is algebraically closed, then this condition implies that the center is a fusion category, so that we recover Muger’s result.

Overgroups of Subsystem Subgroups in Exceptional Groups: Levels

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ H ≤ G(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and there are no roots in Φ orthogonal to all of ∆. There are 72 such pairs (Φ, Δ)$. For F4 and G2 we assume, moreover, that 2 ∈ R * or 6 ∈ R *, respectively. For all such subsystems Δ we construct the levels of intermediate subgroups. We prove that these levels are determined by certain systems of ideals in R, one for each Δ-equivalence class of roots in Φ\∆, and calculate all relations among these ideals, in each case. Bibliography: 64 titles.

The group of commutativity preserving maps on upper triangular matrices over a commutative ring

Let 𝒯 = 𝒯 n (R) be the associative algebra of all n × n upper triangular matrices over a unital commutative ring R with n > 2. A map σ on 𝒯 is called preserving commutativity in both directions if xy = yx ⇔ σ(x)σ(y) = σ(y)σ(x). For an invertible linear map σ on 𝒯, the following two conditions are shown to be equivalent: (a) σ preserves commutativity in both directions and (b) σ takes the form: σ(X) = cS −1[ϵX + (ϵ − 1)PX′P]S + f(X)I, ∀X ∈ 𝒯, where c ∈ R is invertible, ϵ ∈ R is idempotent, i.e. ϵ2 = ϵ, S ∈ 𝒯 is invertible, , X′ means the transpose of X and f is a linear function from 𝒯 to R such that 1 + f(I) is invertible. This result extends the main theorem of Marcoux and Sourour [L.W. Marcoux and A.R. Sourour, Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras, Linear Algebra Appl. 288 (1999), pp. 89–104] to an arbitrary commutative ring.

Module categories for group algebras over commutative rings

AbstractWe develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

Fourth-degree invariants for G(E7,R) not depending on the characteristic

The Chevalley group of type E7 over a field of characteristic different from 2 coincides with the stabilizer of a fourth-degree form on a 56-dimensional vector space. Removing the constraint on the characteristic requires considering asymmetric forms. The space of four-linear forms stabilized by the Chevalley group of type E7 in the minimal representation over an arbitrary commutative ring is described.

Triple and generalized triple derivations of the parabolic subalgebras of the general linear Lie algebra over a commutative ring

Let R be an arbitrary commutative ring with identity and be the general linear Lie algebra over R. Let P be a parabolic subalgebra of . The main aim of this article is to describe triple derivations and generalized triple derivations of P. We show that any triple derivation of P can be uniquely expressed as a sum of standard triple derivations and any generalized triple derivation of P can be expressed as a sum of a scalar map and a triple derivation.

Closed nets in linear groups

The paper reviews the author’s results in the theory of elementary nets (carpets). In particular, closed (admissible) nets are investigated. For an elementary net (a net considered without the diagonal) of additive subgroups of an arbitrary commutative ring, the concepts of the derivative net, the closure of the net, and the net associated with the elementary group are introduced. Factorization of the elementary groups is proposed. This factorization is then used to construct an example of a closed (admissible) net which cannot be completed to a (complete) net. For a third-order elementary net σ of additive subgroups of a commutative ring, decomposition of an elementary transvection from the elementary group E(σ) is obtained.

ON PRIME SUBMODULES AND PRIMARY DECOMPOSITIONS IN TWO-GENERATED FREE MODULES

In this paper, we consider the free R-module R⊕R, where R is an arbitrary commutative ring with identity. We give a full characterization for prime submodules of R⊕R and a useful primeness test for a finitely generated submodule of R⊕R. We study the existence of primary decomposition of a submodule of R⊕R and characterize the minimal primary decomposition. As applications of our results, we give some examples of primary decompositions in R⊕R.

On the Cozero-Divisor Graphs of Commutative Rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.

1-generator quasi-cyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring with \(1\ne 0\). 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let \(\gamma \) be a fixed generator of the maximal ideal of R, \(F=R/\langle \gamma \rangle \) and \(|F|=q\). For any positive integers m, n satisfying \(\mathrm{gcd}(q,n)=1\), let \(\mathcal{R}_n=R[x]/\langle x^n-1\rangle \). Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator \(\mathcal{R}_n\)-submodules of the module \(\mathcal{R}_n^m\). First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that \(\mathrm{gcd}(|q|_n,m)=1\), where \(|q|_n\) denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.

On Families of Graphs of Large Cycle Indicator, Matrices of Large Order and Key Exchange Protocols With Nonlinear Polynomial Maps of Small Degree

This paper studies the group theoretical protocol of Diffie–Hellman key exchange in the case of symmetrical group \({S_{p^n}}\) and more general Cremona group \({C(\mathbb K^n)}\) of polynomial automorphisms of free module \({\mathbb K^n}\) over arbitrary commutative ring \({\mathbb K}\). This algorithm depends very much on the choice of the base \({g_n \in C( \mathbb K^n)}\). It is important to work with the base \({g_n \in C( \mathbb K^n)}\), which is a polynomial map of a small degree and a large order such that the degrees of all powers \({g_n^k}\) are also bounded by a small constant. We suggest fast algorithms for generation of a map \({g_n={f_n} \xi_nf_n^{-1}}\), where ξn is an affine transformation (degree is 1) of a large order and fn is a fixed nonlinear polynomial map in n variables such that \({f_n^{-1}}\) is also a polynomial map and both maps fn and \({f_n^{-1}}\) are of small degrees. The method is based on properties of infinite families of graphs with a large cycle indicator and families of graphs of a large girth in particular. It guaranties that the order of gn is tending to infinity as the dimension n tends to infinity. We propose methods of fast generation of special families of cubical maps fn such that \({f_n^{-1}}\) is also of degree 3 based on properties of families of graphs of a large girth and graphs with a large cycle indicator. At the end we discuss cryptographical applications of maps of the kind τ fn τ−1 and some graph theoretical problems motivated by such applications.

On representation categories of wreath products in non-integral rank

For an arbitrary commutative ring k and t∈k, we construct a 2-functor St which sends a tensor category to a new tensor category. By applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra. This generalizes the construction of Deligne’s category Rep(St,k) for representation categories of symmetric groups.

A Generalization of the Cayley-Hamilton Theorem

Let A = [aij]n × n and B = [bij]n × n be two commuting square matrices of order n over an arbitrary commutative ring. We prove that the determinant of the matrix [bijA − aijB]n × n, which is regarded as an n × n block matrix with pairwise commuting entries, is exactly equal to the n × n zero matrix. If B is the identity matrix, then the result is equivalent to the Cayley-Hamilton theorem.

Well-centered overrings of a commutative ring in pullbacks and trivial extensions

Let $R$ be a commutative ring with identity and $T(R)$ its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of commutative rings satisfying this property and to shed light on some open questions raised by Heinzer and Roitman in \cite{HR}.

K-quasiderivations

Abstract A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss several interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extending a previous result. In particular, we show that any such K-quasiderivation must be linear over R. We then discuss two previously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K-quasiderivations which includes the ordinary derivative as a special case.

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid $\pi_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale $\pi_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.

Hereditarily pure associative algebras over a Dedekind ring whose maximal ideals have finite indices

The concepts of purity and of weak purity play an important part in Abelian group theory. It turns out that these concepts may be approached by using the theory of varieties of groups. The idea behind the approach allowed their analogs to be defined for arbitrary algebras and made it possible to formulate major problems of research in this direction (see [1]). Within these frames, the analogs mentioned were explored for modules [2-4], universal algebras [5-7], semigroups [8-12], Lie algebras over an arbitrary associative commutative ring with unity [13], and associative rings [14]. In so doing, more attention was centered on the concept of atomic purity, an analog of weak purity. In fact, atomic purity seems to be more natural to be dealt with in studying a good many algebras, at least at the initial stage. In the present paper, we will look at this concept as applied to associative algebras over a Dedekind ring. (In our argument, atomic purity will be referred to merely as purity.) Namely, we will work to tackle the problem of describing hereditarily pure algebras [1, Problem 17] for a variety of all algebras over a Dedekind ring R whose maximal ideals have finite indices. (In what follows, unless otherwise stated, we retain the designation R for a ring with the property specified.) Note that our reasoning covers the case of algebras over a finite field and the case of associative rings, and incidentally, refines a result announced in [14]. We give necessary definitions and designations. The letter p always stands for a prime, while k, m, and n are used to denote nonnegative integers. The cardinal number of a set M is denoted by |M |. Throughout the paper, a module is a left unitary R-module; an algebra is an associative