Commutative Ring Research Articles

Extended Total Graph Associated with Finite Commutative Rings

Extended Total Graph Associated with Finite Commutative Rings

Factorization of Modules on Commutative Rings

In this article, the factorization of a torsion module structure is examined and definitions and theorems related to the uniform factorization part in modules are given. Then, the prime sub-modules of the modules that can be factorized by a single method are examined and basic definitions and theorems are given. We have also studied the module elements in written form as the product of the unreducible elements of the ring with the unreducible elements of the modules with the help of weak prime units defined on the torsion modules.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

The First General Zagreb Index of the Zero Divisor Graph for the Ring Zpqk

This study investigates the application of graph theory in analyzing the zero divisor graph of a commutative ring, with a specific focus on its connection to the topological index. For an undirected graph Γ with consists of a non-empty set of vertices, V , and a set of edges, E, the first general Zagreb index is defined as a graph invariant that measures the sum of the degree of each vertex to the power of α= 0. Meanwhile, the zero divisor graph Γ of the commutative ring, R is the (undirected) graph with vertices the zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0. In this paper, the general formulas of the first general Zagreb index of the zero divisor graph for the ring of integers modulo pqk are computed for the cases δ = 1, 2, and 3. This research focuses on the ring defined as the integers modulo pqk, where k is a positive integer, p and q are primes p < q. Two examples are given to demonstrate the main f indings.

How many principal prime ideals are there in a polynomial ring?

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an indeterminate over [Formula: see text], [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the power series ring over [Formula: see text], and [Formula: see text] be a principal prime ideal of [Formula: see text] with [Formula: see text]. It is well known that if [Formula: see text] is an integral domain, then [Formula: see text] is a DVR and [Formula: see text] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) [Formula: see text] is a field, (ii) [Formula: see text] is a DVR, but (iii) there is a ring [Formula: see text] such that [Formula: see text] has no principal prime ideal. We also study the maximal ideals of [Formula: see text] that are principal.

Fermat's Lemma on the Hyperbolic Plane

Generalizing classical mathematical problems to other spaces is an important research direction in mathematics. This paper studies Fermat's Lemma on the hyperbolic plane, where the set of hyperbolic numbers is a commutative ring with zero divisors generated by two real numbers, similar to the complex numbers but more complicated. Compared to the classical Fermat’s Lemma, the Fermat’s Lemma in this paper has a higher dimension and is built on a hyperbolic ring whose algebraic structure is more complex than the real number field. On the basis of a generalization of the classical Fermat’s Lemma, the paper overcomes the difficulty that hyperbolic numbers contain zero divisors, and obtains the proof of the first and second Fermat’s Lemma on the hyperbolic plane through the decomposition of hyperbolic numbers, which will give impetus to theoretical research in hyperbolic analysis.

Taylor's formula on the separating complex plane and its applications

This article begins by exploring split-complex numbers, also known as dual or double numbers, a mathematical concept that extends the real number system by creating a commutative ring with a zero divisor. Furthermore, the paper extends the Taylor series theory from the real analysis domain to the domain of split-complex numbers, thereby establishing the corresponding Taylor formula on the split-complex plane. The introduction of split-complex numbers opens up new perspectives and provides new tools for the modeling and analysis of quantum systems. In the field of electromagnetism, the application of split-complex numbers also demonstrates its potential, contributing to a more precise description of the behavior of electromagnetic fields under specific conditions. These theoretical advancements are expected to further promote research in the field of split-complex analysis, playing a more critical role in important branches of physics such as quantum mechanics and electromagnetism, thereby facilitating the development of related theories and the emergence of innovative practical applications.

The First Mean Value Theorem of integral in Perplex Plane

Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.

Rouquier dimension versus global dimension

We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.

Exploring the zero-divisor graph over commutative ring: topological examine of algebraic structure

Exploring the zero-divisor graph over commutative ring: topological examine of algebraic structure

Integral closure of an affine algebra

Let R be a commutative ring with identity and R ′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R ′ is finitely generated, i.e., R ′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R ≤ 2 , then R ′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R = ∞ or reg-dim R = n ≤ dim R = n + m − 1 for arbitrary positive integers n, m.

Torsion-simple objects in abelian categories

We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Enumeration of matrices with single unit-entry rows over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, formal expressions of the number of [Formula: see text] matrices over [Formula: see text] of rank [Formula: see text] and the number of invertible matrices over [Formula: see text] are presented. The number of matrices over [Formula: see text] with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

When quotient of a polynomial or power series ring by a monomial ideal is half-factorial

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] . We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R [ x ] or R [ [ x ] ] is half-factorial for an arbitrary commutative ring R, then R is an integral domain.

Rigid integral representations of quivers over arbitrary commutative rings

Rigid integral representations of quivers over arbitrary commutative rings

Characteristic modules and Gorenstein (co-)homological dimension of groups

In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded Z-valued functions on such a group G to characterize the Gorenstein flat ZG-modules, in terms of flat modules, and the Gorenstein injective ZG-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective ZG-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective ZG-module is Gorenstein flat and (b) a ZG-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.

On gr-n-submodules of graded modules over graded commutative rings

Let G be a group with identity e. Let ℜ be a G-graded commutative ring, ℑ be a graded ℜ -module. In this paper, we introduce and study the concept of graded n-submodules of ℑ. We obtain many results concerning graded n-submodules. Some characterizations of graded n-submodules and their homogeneous components are given. A proper graded submodule U of ℑ is said to be a graded n-submodule if whenever r ∈ h(ℜ), m ∈ h(ℑ) with rm ∈ U and r ∉ Gr(Annℜ(ℑ)), then m ∈ U.