Commutative Ring Research Articles

Codimension One Foliation and the Prime Spectrum of a Ring

Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M. A leaf class of a leaf F is the union of all leaves having the same closure as F . Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [2, Theorem 3.15] it is shown that if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the prime spectrum (or simply the spectrum) of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below.

Lagrange's mean value theorem in hyperbolic number plane

Hyperbolic number is a generalization of real number, it is a commutative ring of zero factor, and has been widely used in many fields such as mathematics, physics and engineering. In this paper, we first review the basic concepts and properties of hyperbolic numbers, and then study the Lagrange's mean value theorem on the hyperbolic numbers plane, and get a more general theorem. This not only provides impetus for the development of hyperbolic analysis, but also further injects energy into the development of applied mathematics.

Schwarz's lemma on the perplex plane

Perplex number is an extension of complex numbers, constituting a commutative ring with zero divisors. In this paper, we primarily investigate the Schwarz lemma on the perplex plane and successfully obtain an estimation of the modulus of differentiable functions in perplex plane. Furthermore, we discuss its potential applications in complex analysis and geometric function theory. This conclusion not only offers a fresh perspective for mathematical research but also holds significant reference value for advancing studies in physics, providing valuable theoretical support for subsequent scientific investigations.

On the comaximal graph of a ring

In this paper, we prove that the study of the comaximal graph of a finite commutative ring with identity is the same as the study of the zero-divisor graph of the specially constructed lattice. Also, we introduce the blow-up of a Boolean lattice using finite chains. Furthermore, we prove that in the case of a reduced ring, the comaximal graph and the co-annihilating graph coincide. In the last section, we study the perfectness of the comaximal graphs and characterize the Hamiltonian property of the complement of a comaximal graph.

G-Small Intersection Graph of a Module

Let be a commutative ring with identity, and be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of are studied. Properties of such as connectedness, and completeness are considered. Besides, the girth and the diameter of are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph is complete if, is a generalized hollow module or is a direct sum of two simple modules, is proved.

Linear complementary pairs of constacyclic n-D codes over a finite commutative ring

Linear complementary pairs of constacyclic n-D codes over a finite commutative ring

A look at FPF rings

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring [Formula: see text] is reduced FPF if and only if it is [Formula: see text]-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a [Formula: see text] to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space [Formula: see text] is fraction-dense if and only if [Formula: see text] is a continuous ring.

Self-covering, finiteness, and fibering over a circle

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold M M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group Z \mathbb Z is a fiber bundle over S 1 S^1 , except for the 4 4 -dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with nonfree abelian fundamental group, which are not fiber bundles over S 1 S^1 .

Anti-isomorphism between Brauer groups BQ(S, H) AND BQ(Sop , H∗)

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel’d H-module algebras and the Brauer group BQ ( R , H * ) of Hopf Yetter-Drinfel’d H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel’d (S, H)-module algebras and BQ ( S o p , H * ) , the Brauer group of dyslectic Hopf Yetter-Drinfel’d ( S o p , H * ) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel’d H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque

A refined scissors congruence group and the third homology of SL2

There is a natural connection between the third homology of SL2(A) and the refined Bloch group RB(A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE2-domain such that −1∈A×2, then we have the exact sequenceH3(SM2(A),Z)→H3(SL2(A),Z)→RB(A)→0, where SM2(A) is the group of monomial matrices in SL2(A). Moreover, we show that RP1(A), the refined scissors congruence group of A, is naturally isomorphic to the relative homology group H3(SL2(A),SM2(A);Z).

Some properties of generalized comaximal graph of commutative ring

Some properties of generalized comaximal graph of commutative ring

Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree diagonal matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. The cozero-divisor graph [Formula: see text] of a finite commutative ring [Formula: see text] with unity is a simple undirected graph with the set of all nonzero nonunits of [Formula: see text] as vertices and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, we study structural properties of [Formula: see text] by defining an equivalence relation on its vertex set in terms of principal ideals of the ring [Formula: see text]. Then we obtain the universal adjacency eigenpairs of [Formula: see text] and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of [Formula: see text] and [Formula: see text] in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo [Formula: see text] in a simpler form.

Some Properties of S-Semiannihilator Small Submodules and S-Small Submodules with respect to a Submodule

Let R be a commutative ring with nonzero identity, S⊆R be a multiplicatively closed subset of R, and M be a unital R-module. In this article, we introduce the concepts of S-semiannihilator small submodules and S-T-small submodules as generalizations of S-small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.

Applications of reduced and coreduced modules I

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

Automorphisms of Chevalley groups over commutative rings

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.

On S C D F -Modules

A module M over a commutative ring is termed an SCDF-module if every Dedekind finite object in σ[M] is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of SCDF-modules. These include local SCDF-modules, finitely generated $SCDF$-modules, and hollow SCDF-modules with Rad(M)=0≠M. Additionally, we examine QF SCDF-odules in the context of duo-ring.

On degree-based entropy measure for zero-divisor graphs

Let [Formula: see text] be a commutative ring with [Formula: see text] A zero-divisor graph [Formula: see text] is an undirected graph on the vertex set [Formula: see text] such that any two nonzero elements [Formula: see text] are adjacent whenever [Formula: see text] The concept of entropy measure is fundamental and has been the subject of interest in various fields. The purpose of this paper is to calculate the entropy measure of [Formula: see text] for various rings. We take into consideration the ring [Formula: see text] for [Formula: see text] where [Formula: see text] and [Formula: see text] are distinct prime numbers. Moreover, we also discussed the entropy measure of the zero-divisor graph for the rings [Formula: see text] and [Formula: see text] This study will help us further understand the algebraic structure of the commutative rings.

Invertible bases and root vectors for analytic matrix-valued functions

We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring $R$. We first review the conditions for the existence of a basis for submodules of $R^n$ where $R$ is a Bézout domain. Then, we define the concept of invertible basis of a submodule of $R^n,$ and when $R$ is an elementary divisor domain, we link it to the Main Theorem of G. D. Forney Jr. [SIAM J. Control, 13:493-520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of $R^n$. As an application, we let $\Omega \subseteq \mathbb{C}$ be either a connected compact set or a connected open set, and we specialize to $R=\mathcal{A}(\Omega)$, the ring of functions that are analytic on $\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}(\Omega)^{m \times n}$, $\ker A(z) \cap \mathcal{A}(\Omega)^n$ is a free $\mathcal{A}(\Omega)$-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we use invertible bases to define and study maximal sets of root vectors at $\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.

Differential graded algebras for trivalent plane graphs and their representations

To any trivalent plane graph embedded in the sphere Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank r representations of this dg-algebra, over a field \mathbb{F} , correspond to colorings of the faces of the graph by elements of the Grassmannian \operatorname{Gr}(r,2r;\mathbb{F}) so that bordering faces are transverse, up to the natural action of \operatorname{PGL}_{2r}(\mathbb{F}) . Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that, for Legendrian 2-weaves, rank r representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank r . This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.

Characterizing Rickart and Baer ultragraph Leavitt path algebras

We characterize ultragraph Leavitt path algebras that are Rickart, locally Rickart, graded Rickart, and graded Rickart *-rings. We also characterize ultragraph Leavitt path algebras that are Baer, locally Baer, graded Baer, Baer *-rings, and combinations of these. These characterizations build on and generalize the work of Hazrat and Vaš on Leavitt path algebras over fields to ultragraph Leavitt path algebras over semi-simple commutative unital rings.