Ring Theory

We give a description of a 2-torsion free Vinberg (−1, 1) ring R. If every nonzero root space of R − for S is one-dimensional where S is a split abelian Cartan subring of R − which is nil on R then R is a Lie ring isomorphic to R −. This generalizes the known result obtained by Myung for the case that R is a 2-torsion free Vinberg (−1, 1) ring and is power associative. We also give a condition that a Levi factor C of R − be an ideal of R when the solvable radical of R − is nilpotent. We apply these results for reductive case of R − .

Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Futhermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

Ring theory is one of the branches of the abstract algebra that has been broadly used in images. However, ring theory has not been very related with image segmentation. In this paper, we propose a new index of similarity among images using rings and the entropy function. This new index was applied as a new stopping criterion to the Mean Shift Iterative Algorithm with the goal to reach a better segmentation. An analysis on the performance of the algorithm with this new stopping criterion is carried out. The obtained results proved that the new index is a suitable tool to compare images.

Ring Theory states that a ring is an algebraic structure where two binary operations can be performed within the elements: addition and multiplication. Image segmentation is the process of partitioning an image into pixels and as- signing each pixel a value. Binary images are the result of colored pictures being converted into gray-scale and individual pixels being assigned a value between 0 and 1 based on the presence of darkness. The Mean Shift Iterative Algorithm is a method of image segmentation that arranges pixels into clusters that share the same average pixel value. This paper examines the pre-existing method of clustering and incorporates the concept of Ring Theory to improve the process of converting grayscale images into binary resolutions for implementation into medical technology.

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

(-1, 1) ring satisfying the Weak Novikov identity (w, x, yz) = y(w, x, z) is studied in this paper. The ring is associative under the following assumption that (i) it has finite number of generators, (ii) it is a 2-torsion free prime ring and has minimum condition on either right ideals or on trivial left ideals,(iii) is a 2-torsion free simple and (iv)is a semiprime ring which is both (-1, 1) and (1, 0).

In this talk several properties of the inverse along an element in the context of unitary rings will be presented. Among others results, the set of all invertible elements along a fixed element will be fully described. In addition, commuting inverse along a fixed element will be characterized. Furthermore, the special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered.

Este texto está dividido en dos partes principales: anillos y campos. La primera parte, sobre anillos, consta de ocho capítulos. En los primeros dos se definen varios tipos de anillos, como los dominios enteros, y se abordan conceptos básicos relacionados con los anillos. En los capítulos 3, 4 y 5 se desarrolla parte de la teoría elemental de los anillos, tratando temas como los ideales, los anillos cociente y los homomorfismos. En los capítulos 6 y 7 se presenta un anillo de particular importancia: el anillo de polinomios, el cual será fundamental para el desarrollo posterior del texto. Finalmente, en el capítulo 8 se estudian tres tipos de dominios enteros especiales: los dominios de ideales principales, los dominios de factorización única y los dominios euclidianos.
La segunda parte, sobre campos, consta de cinco capítulos. En los capítulos 9, 10 y 11 se desarrolla la teoría de campos clásica, con enfoque principalmente en las extensiones de campos. Los últimos dos capítulos, el 12 sobre campos finitos y el 13 sobre teoría de Galois, están escritos de tal forma que sean independientes uno del otro y cualquiera de los dos es una buena culminación para un curso de álgebra moderna.

Let F q [ε] := F q [X]/(X 4 − X 3) be a finite quotient ring where ε 4 = ε 3 , with F q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over F q [ε], ε 4 = ε 3 of characteristic p = 2, 3 given by homogeneous Weierstrass equation of the form Y 2 Z = X 3 + aXZ 2 + bZ 3 where a and b are parameters taken in F q [ε]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve E a,b (F q [ε]) and we will show that E π0(a),π0(b) (F q) and E π1(a),π1(b) (F q) are two elliptic curves over the finite field F q , such that π 0 is a canonical projection and π 1 is a sum projection of coordinate of element in F q [ε]. Precisely, we give a classification of elements in elliptic curve over the finite ring F q [ε].

Despite its relative brevity, Paul's first letter to the Thessalonians has posed several obstacles to interpreters, including the presence of two separate thanksgivings (1:1-10; 2:13-16?), stylistic shifts (3:9-13; 5:1-3), a premature clos­ing (4:1-2), and the placement of an eschatological itinerary (4:13-5:10) between two sets of moral exhortations (4:1-12; 5:12-24). These structural problems pre­cipitate into the larger exegetical question of whether the delimitation of the letter can communicate any insights on the author's intended message. That is, what is the connection between the letter's form and rhetorical function? In this article I will argue that the letter consists of a complex A, B, A ' structure of three ring sets (1:2-2:13; 2:14-3:13; 4:1-5:26), each of which comprises three or more units. In contrast to previous analyses of the letter that have relied on subjective criteria, the structure herein is based on lexico-grammatical evidence and the cultural milieu of orality in late antiquity. When the letter is viewed as an oratorical per­formance of complex concentric patterns, one can trace several rhetorical lines of thought to benefit the letter's interpretation.

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.

This book engages the structure and message of 1 Corinthians within its most relevant context of late Western antiquity's oral culture. Using a text-centered methodology, Timothy Milinovich demonstrates and analyzes a series of concentric patterns (or ring formations) through which Paul develops his arguments to the Corinthian church. Such patterns were ubiquitous in oral cultures and their literature. These structures, which are defined by objective lexical repetitions, aid the interpretation of an overall concentric pattern of three sections (A, 1:1--4:21; B, 5:1--11:1; A´, 11:2--16:24), nine ring sets (a, 1:1-17; b, 1:18--3:3; a´, 3:4--4:21; a, 5:1--6:20; b, 7:1-40; a´, 8:1--11:1; a, 11:2--14:40; b, 15:1-58; a´, 16:1-24), thirty-five ring units (e.g., 5:1-13; 10:1-17; 15:12-24), and numerous micro-rings (e.g., 4:6-8; 8:1-4). Analyzing these lexical repetitions presents a demonstrably coherent message as it progresses through the concentric portions of the text.

These findings represent a departure from previous treatments of the letter as if it were a modern, linear essay. As shown throughout this work, many linear treatments view the units like wooden blocks, only to build a single, unbalanced tower, and thus can miss important rhetorical connections in the concentric textual units. Milinovich treats the units and sets like interlocking pieces to present the inherent cohesiveness of the complex yet integral exhortation to grace, love, and unity that Paul wished to convey to this community on the verge of collapse.

Ring theory is one of the branches of the abstract algebra that has been broadly used in images. However, ring theory has not been very related with image segmentation. In this paper, we propose a new index of similarity among images using Zn rings and the entropy function. This new index was applied as a new stopping criterion to the Mean Shift Iterative Algorithm with the goal to reach a better segmentation. An analysis on the performance of the algorithm with this new stopping criterion is carried out. The obtained results proved that the new index is a suitable tool to compare images.

Ring theory is one of the branches of the abstract algebra that has been broadly used in images. However, ring theory has not been very related with image segmentation. In this paper, we propose a new index of similarity among images using ℤn rings and the entropy function. This new index was applied as a new stopping criterion to the Mean Shift Iterative Algorithm with the goal to reach a better segmentation. An analysis on the performance of the algorithm with this new stopping criterion is carried out. The obtained results proved that the new index is a suitable tool to compare images.

The converse of Schur's lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.

Let R and R be two unital rings such that R contains a non-trivial idempotent P 1. If R is a prime ring, we characterize the form of bijective map ϕ : R → R which satisfies ϕ(ABP) = ϕ(A)ϕ(B)ϕ(P), for every A, B ∈ R and P ∈ {P 1 , P 2 }, where P 2 := I − P 1 and I is the unit member of R. It is shown that ϕ is an isomorphism multiplied by a central element. Finally, we characterize the form of ϕ : R → R which satisfies ϕ(P)ϕ(A)ϕ(P) = P AP , for every A

A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A#H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that the Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant  in case A is noetherian, connected graded, and Koszul.

In this paper we consider prime graph of R (denoted by ) of an associative ring R (introduced by Satyanarayana, Syam Prasad and Nagaraju [6]). This short paper is divided into two Sections. Section-1 is devoted for preliminary definitions. In section-2, we constructed Left zero divisor graph of R (denoted by LZDG(R)) where R = the set of all matrices over the ring of integers modulo 2.

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.

A commutative loop or ring is said to be Jordan if it satisfies the
identity $(x^2y)x = x^2(yx)$. We show that the loop ring of a Jordan loop L is Jordan and not associative only if the characteristic of the coefficient ring is even and call such a loop ring Jordan (RJ, for short). While Jordan loops are in general not power associative, RJ loops are. We give various constructions of finite RJ loops and conjecture that these exist only when they have order divisible by four. We also conjecture that RJ loops are
precisely those commutative loops in which squares are in the left nucleus.

The existence of loop rings that are not associative but which satisfy the Moufang or Bol identities is well known. Here we complete work started 25 years ago by establishing the
existence of loop rings that satisfy any identity of “Bol–Moufang” type (without being associative). As it turns out, with one exception, loop rings satisfying an identity of Bol–Moufang type all satisfy a Moufang or Bol identity. We also highlight some similarities and differences in the consequences of several Bol–Moufang identities as they apply to loops and rings.

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.

In this article, we introduce new generators of a permuting n-derivations to improve and increase the action of usual derivation. We produce a permuting n-generalized semiderivation, a permuting n-semigeneralized semiderivation, a permuting n-antisemigeneralized semiderivation and a permuting skew n-antisemigeneralized semiderivation of non-empty rings with their applications. Actually, we study the behaviour of those types and present their results of semiprime ring R. Examples of various results have also been included. That is, many of the branches of science such as business, engineering and quantum physics, which used a derivation, have the opportunity to invest them in solving their problems.

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.

We characterize the diagonalizable subalgebras of End(V ), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the "finite topology") on End(V ), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V ). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn-Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

An extensive generalized concept of classical ring set forth the notion of a gamma ring theory. As an emerging field of research, the research work of classical ring theory to the gamma ring theory has been drawn interest of many algebraists and prominent mathematicians over the world to determine many basic properties of gamma ring and to enrich the world of algebra. The different researchers on this field have been doing a significant contributions to this field from its inception. In recent years, a large number of researchers are engaged to increase the efficacy of the results of gamma ring theory over the world.

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.

Let R be an associative ring. We define a subset S R of R as S R = {a ∈ R | aRa = (0)} and call it the source of semiprimeness of R. We first examine some basic properties of the subset S R in any ring R, and then define the notions such as R being a |S R |-reduced ring, a |S R |-domain and a |S R |-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite |S R |-domain is necessarily unitary, and is in fact a |S R |-division ring. However, we provide an example showing that a finite |S R |-division ring does not need to be commutative. All possible values for characteristics of unitary |S R |-reduced rings and |S R |-domains are also determined.

Let $R$ be a ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a
graph whose vertices are all non-trivial left ideals of $R$, and two distinct vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$.
In this paper, some results on the co-maximal ideal graphs of matrix algebras are given. For instance, we determine the domination number, the clique number and a lower bound of the independence number of $\Gamma(M_n(\mathbb{F}_q))$. Furthermore, we characterize all rings (not necessarily commutative) whose domination numbers of their co-maximal ideal graphs are finite. Among other results, we show that  if $\Gamma(R)\cong \Gamma(M_n(\mathbb{F}_q))$,
where $n\geq 2$ is a positive integer and $R$ is a ring, then $R\cong M_n(\mathbb{F}_q)$.
Also, it is proved that if $R$ and $R'$ are two finite reduced rings and $\Gamma(M_m(R))\cong \Gamma(M_n(R'))$,
for some positive integers $m,n\geq 2$, then $m=n$ and $R\cong R'$.