Ring Theory Research Articles

A note on generalized m-derivations to weakly cancellative semirings

Objectives: Semirings is an important algebraic structure with applications in theory of automata, formal languages and theoretical computer science. The mappings which enforces commutativity in semirings remains attractive for researchers, since commutativity would be helpful in calculations and bring it\'s applications to ease. Our aim is to enforce commutativity in semirings by generalizing the classical theorem of Martindale [14, Theorem 3] with generalized m-derivation. Further, we discuss that composition of two generalized m-derivations ensure that one of their associated derivation must be trivial. Method: We use generalized m-derivations which is associated to multiplicative derivations in certain semirings. Findings: We find the conditions of commutativity in semirings through these particular generalized m-derivations. Moreover, we discuss the characteristics of these mappings in weakly cancellative semirings. Novelty: The concept of generalized m-derivations is newly introduced by us in ring theory in (1) and here we extend this concept to theory of semirings. We attempt to induce commutativity in weakly cancellative semirings (2) whose concept is unorthodox in the theory of semirings. This article pave new ways to study derivations and its applications on semirings. Keywords: Derivations; generalized m-derivations; weakly cancellative semiring; commutativity

Bivariant derived algebraic cobordism

We extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraicK0K^0analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.

On near generalized rings

Near ring is a generalization of ring: addition need not beabelian, and (more importantly) only one distributive law is postulated. In this paper, we establish a new generalization of near ring, namely near generalized ring. We also investigate some interesting algebraic properties of the near generalized ring. Moreover, we discuss the concept of commutativities and homomorphisms of near generalized rings and then some related properties are investigated. Furthermore, we introduce the notion of quotient structures of near generalized rings using congruence relations. Finally, fundamental theorem of near generalized ring homomorphisms analogues of ring theory is obtained.

Computing the size of zero divisor graphs

A recent subject of study linking commutative ring theory with graph theory has been the concept of the zero divisor graph of a commutative ring. The zero divisor graph of a commutative ring exhibits a remarkable amount of graphical structure. Let G(R) be the zero divisor graph introduced by Beck [9], whose vertices are the elements of a ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Let Г(R) be the zero divisor graph introduced by Anderson, Livingston [5] whose vertices are the non-zero zero divisors of the ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Here, the authors investigate the size of the graphs G(ℤ n ), Г(ℤ n ).

Dual of Extending Acts

Since 1980s, the study of the extending module in the module theory has been a major area of research interest in the ring theory and it has been studied recently by several authors, among them N.V. Dung, D.V. Huyn, P.F. Smith and R. Wisbauer. Because the act theory signifies a generalization of the module theory, the author studied in 2017 the class of extending acts which are referred to as a generalization of quasi-injective acts. The importance of the extending acts motivated us to study a dual of this concept, named the coextending act. An S-act MS is referred to as coextending act if every coclosed subact of Ms is a retract of MS where a subact AS of MS is said to be coclosed in MS if whenever the Rees factor ⁄ is small in the Rees factor ⁄then AS=BS for each subact BS of AS. Various properties of this class of acts have been examined. Characterization of this concept is intended to show the behavior of a coextending property. In addition, based on the results obtained by us, the conditions under which subacts inherit a coextending property were demonstrated. Ultimately, a part of this paper

On Mixed Fuzzy Topological Ring

The theory of fuzzy topological ring has wide scope of applicability than order topological ring theory. The reason is fuzzy can provide better result. Therefore, fuzzy topological ring has been found in Robotics, computer, artificial intelligent, etc. In this paper, we induce mixed fuzzy topological ring space and fuzzy neighborhoods system of mixed fuzzy topological ring space. Also, we study fuzzy continuity of mixed fuzzy topological ring.

On modules with self Tor vanishing

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of the vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory.

Approximations of maximal and principal prime Ideal

In this paper, we will delving deeper into the connection between the rough theory and the ring theory precisely in the maximal ideal. We will find that there is a rough maximal and principal ideal as an extension of the notion of a maximal and principle ideal respectively. Some properties of approximations of the maximal and the principal are studied.

On weakly local rings

This article concerns a property of local rings and domains. A ring $R$ is called weakly local if for every $a\in R$, $a$ is regular or $1-a$ is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

Congruence classes and extensions of rings with an application to braces

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring [Formula: see text] by an ideal [Formula: see text] is a paragon in the truss [Formula: see text] associated to [Formula: see text]. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated to a ring, the resulting object is a truss, never a ring, unless the module is trivial. On the other hand, if the extended truss is associated to a brace, the resulting truss is also associated to a brace, irrespective of the module used.

Boolean lifting property in quantales

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

Lensing by Kerr black holes

Interpreting horizon-scale observations of astrophysical black holes demands a general understanding of null geodesics in the Kerr spacetime. These may be divided into two classes: "direct" rays that primarily determine the observational appearance of a given source, and highly bent rays that produce a nested sequence of exponentially demagnified images of the main emission: the so-called "photon ring". We develop heuristics that characterize the direct rays and study the highly bent geodesics analytically. We define three critical parameters $\gamma$, $\delta$, and $\tau$ that respectively control the demagnification, rotation, and time delay of successive images of the source, thereby providing an analytic theory of the photon ring. These observable parameters encode universal effects of general relativity, independent of the details of the emitting matter.

Error analysis of undergraduate students in solving problems on ring theory

This study aimed to describe the types of error of undergraduate students in solving problems on ring theory. It involved 46 undergraduate students of Mathematics Education, Faculty of Teacher Training and Education, Sebelas Maret University academic year of 2018/2019. The research type used in this study was qualitative with a case study design. Data was gathered through tests and interviews. Validity tests of the data were a triangulation of methods and using reference materials. The subjects were chosen by purposive sampling technique. Consequently, three students selected to be interviewed. Errors found in this study included translation error, concept error, strategy error, and an encoding error. On translation errors, students could not understand the definition of the set and operations used in the questions. While concept and strategy errors were due to student’s inability to understand and use mathematical definitions, such as the definitions of group, monoid, ring, and commutative ring. Students were unable to write mathematical definitions in the form of standard mathematical symbols, which indicated they did not understand them well. On encoding errors, students only focused on the final answers or conclusions without having proper reasons.

Algebraic properties of expectation semirings

In this paper, we investigate the algebraic properties of the expectation semirings which are semiring version of the concept of trivial extension in ring theory. We discuss ideals, primes, maximals and primary ideals of these semirings. We also discuss the distinguished elements such as the units, idempotents, and zero-divisors of the expectations semirings. Similar to their counterparts in ring theory, we introduce pr\'{e}simplifiable, domainlike, clean, almost clean, and weakly clean semiring and see when an expectation semiring is one of these semirings.

On Jordan Ideals of Inverse Semirings with Involution

Objectives: The main objective of this article is to introduce *-Jordan ideals of a certain class of semirings called MA-semirings with involution and to investigate some conditions for which the above said ideals contained in the center. Methods and findings: We use the Jacobian identities and 2-torsion freeness of MA semirings. In this connection, we establish some important results of ring theory for the class of MA-semirings. Applications/ improvements: The commutative property is helpful to study the theory of semirings with ease therefore we find some conditions to impose commutativity in semirings, which are indeed novel idea in the field of semirings. Furthermore, these conditions are used in a most generalized way that these conditions bring the *-Jordan ideals to the center, therefore, it would be the corollary of result that semiring is commutative. Keywords: Semirings, *-Semirings, MA-Semirings, *-Jordan Ideals, *-Prime Semirings, Involution Mathematics Subject Classifications (2010): 16Y60, 16W10

The Dual Baer Criterion for non-perfect rings

Abstract Baer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.

Ring endomorphisms satisfying the central reversible property

A ring R is called reversible if for $$a, b \in R$$ , $$ab=0$$ implies $$ba=0$$ . These rings play an important role in the study of noncommutative ring theory. Kafkas et al. (Algebra Discrete Math. 12 (2011) 72–84) generalized the notion of reversible rings to central reversible rings. In this paper, we extend the notion of central reversibility of rings to ring endomorphisms. We investigate various properties of these rings and answer relevant questions that arise naturally in the process of development of these rings, and as a consequence many new results related to central reversible rings are also obtained as corollaries to our results.

Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂d, where ∂ is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h∂d. All of the above applies to the theory of formal differential operator rings.

Exact Solutions of Two Nonlinear Partial Differential Equations by the First Integral Method

In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software; the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.