Ring Theory Research Articles

A high-order model for in-plane vibrations of rotating rings on elastic foundation

A new high-order model for in-plane vibrations of rotating rings is developed in this paper. The inner surface of the ring is connected to an immovable axis through an elastic foundation (distributed springs), whereas the outer surface is traction free. The developed model enables the dynamic analysis of the rings on stiff elastic foundation that rotate with a high speed. The traction force at the inner surface of such rings is so high that it influences significantly the through-thickness stress distribution. This boundary effect cannot be captured by the classical low order theories while the model proposed in this paper can account for this effect. Nonlinear equations of motion are first derived, considering the geometrical nonlinearity of the system while assuming the linear elastic behaviour of the ring material. The formulation accounts for the stress caused by rotation and the significant normal and tangential traction forces at the inner surface of the ring. The displacement fields are assumed to be polynomials of the through-thickness coordinate in both the radial and circumferential directions. The derivation is generic and can yield ring theories of different order, i.e. of the Timoshenko-type and beyond, with proper consideration of both the internal state of the body and the boundary effects at the surfaces. Two types of critical speeds are investigated, namely the one at which the free vibrations become unstable and the one at which the forced vibration of a rotating ring subjected to a constant stationary point load experiences resonance. A comparison is presented of the predictions of the developed model to those of the lower order theories. It is shown that even for thin rings on elastic foundation, high order corrections, beyond the ones of the Timoshenko theory, need to be considered for an accurate estimation of the critical speeds of rotating rings. The new high-order model is superior to the existing ring models in predicting dynamic behaviour of either stationary or rotating rings. Without loss of generality, the model is applicable to both plane strain and plane stress configurations.

INTERPRETING ARITHMETIC IN THE FIRST-ORDER THEORY OF ADDITION AND COPRIMALITY OF POLYNOMIAL RINGS

AbstractWe study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, + , \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself.

On a ring property related to nilradicals

In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the K$\ddot{\text{o}}$the's conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.

A completely new approach for the theory of Soft Groups and Soft Rings

A completely new approach for the theory of Soft Groups and Soft Rings

Going up and lying over in congruence-modular algebras

Abstract In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence-modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide algebraic and topological characterizations for them. We also point out many kinds of varieties in which these properties always hold, generalizing the results of Belluce on MV-algebras and Rasouli and Davvaz on BL-algebras.

On domains with properties weaker than atomicity

In general, ring theory is focused on atomic rings, i.e., rings in which every non-zero non-unit element has a factorization into irreducible elements. In a recent article of Boynton and Coykendall, the two authors introduce two properties that are slightly weaker than atomicity, which they call “almost atomicity” and “quasi-atomicity”. In this article, we classify various properties weaker than atomicity and find domains that possess certain combinations of them.

Narain Gupta's three normal subgroup problem and group homology

This paper is about application of various homological methods to classical problems in the theory of group rings. It is shown that the third homology of groups plays a key role in Narain Gupta's three normal subgroup problem. For a free group F and its normal subgroups R,S,T, and the corresponding ideals in the integral group ring Z[F], r=(R−1)Z[F],s=(S−1)Z[F],t=(T−1)Z[F], a complete description of the normal subgroup F∩(1+rst) is given, provided R⊆T and the third, fourth and fifth homology groups of R/R∩S are torsion groups.

A fresh look into monoid rings and formal power series rings

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

Identification of proofs via syzygies

In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.

$K$-Theory of Non-Archimedean Rings. I

We introduce a variant of homotopy K -theory for Tate rings, which we call analytic K -theory . It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K -theory with continuous K -theory, which is defined in terms of models. Along the way we also prove some results about the algebraic K -theory of Tate rings.

Using Divisor Function and Euler Product Function in Abstract Algebra Concepts

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory. In this paper for the most part centered around number theory ideas which are utilized in different themes like group theory and ring theory, these speculations are extremely unique ideas to comprehend among this we might want to express our perspectives as far as number hypothesis/theory ideas, such as, to calculate some subgroups of a cyclic group, number of ideals, principal ideals of a ring and number of generators of a cyclic group as far as both regular procedure and number speculation/hypothesis thoughts.

Algebraic $K$-theory of triangular rings and its generalization

В работе строится "тензорное" обобщение алгебраической $K$-теории верхнетреугольных колец и доказывается, что соответствующие $K_m$-группы естественно изоморфны прямой сумме $K_m$-групп диагональной части. Библиография: 6 названий

ON TORUS ACTIONS OF HIGHER COMPLEXITY

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.

Reflexive-nilpotents-property skewed by ring endomorphisms

The reflexive property for rings was introduced by Mason and play roles in noncommutative ring theory. A ring R is called reflexive if for a, b in R, aRb = 0 implies bRa = 0. Recently, Kheradmand et al. introduced the notion of RNP (reflexive-nilpotents-property) rings by restricting the reflexive property to nilpotent elements. In this article, we study reflexive-nilpotents-property skewed by a ring endomorphism alpha and introduce the notion of alpha -skew RNP rings. We investigate various properties and extensions of these rings and also determine the structure of minimal noncommutative alpha -skew RNP rings.

Forms of Choice in Ring Theory

We investigate the relationship between various choice principles and nhbox {th}-root functions in rings. For example, we show that the Axiom of Choice is equivalent to the statement that every ring has a square-root function. Furthermore, we introduce a choice principle which implies that every integral domain has an nhbox {th}-root function (for odd integers n), and introduce another choice principle which is equivalent to the Prime Ideal Theorem restricted to certain ideals. Finally, we investigate the dependencies between the two new choice principles and a choice principle for families of n-element sets.

Rings and Modules Whose Socles are Relative Ejective

Lifting homomorphism from modules to modules or even from certain submodule to the modules have been important both in ring and module theory. In this note we study rings and modules whose socles are relative ejective. Moreover we reduce our consideration to rings and modules with injective socles which provides the dual notion to PSmodules.

Zariski Topology

Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h −1(𝔭) where 𝔭 2 Spec B.

Idempotents, group membership and their applications

AbstractŠ. Schwarz in his paper [SCHWARZ, Š.:Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.:Contribution to the theory of torsion semigroups, Czechoslovak Math. J.3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.

Exact contexts, noncommutative tensor products, and universal localizations

Exact contexts and their noncommutative tensor products are introduced which generalize the notions of Milnor squares and usual tensor products over commutative rings, respectively. Exact contexts are characterized by rigid morphisms which exist abundantly, while noncommutative tensor products not only capture some useful constructions in ring theory (such as coproducts of rings and trivially twisted extensions) but also provide a new method to construct universal localizations with rich homological and structural information. Moreover, sufficient and necessary conditions in terms of the data of exact contexts are presented to ensure that the universal localizations constructed are homological ring epimorphisms.

The computability, definability, and proof theory of Artinian rings

We show that, in the context of Reverse Mathematics, WKL0 (Weak König's Lemma) implies the statement ART0 which says that every Artinian ring is Noetherian, over RCA0 (Recursive Comprehension Axiom). To achieve this goal, we prove a general Computable Full Structure Theorem for computable Artinian rings similar to the classical version found in most Algebra texts.