Mathematics › Algebra and Number Theory

Rings, Modules, and Algebras

Works Count: 47419

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This cluster of papers focuses on the study of properties and structures of commutative rings, including topics such as zero-divisor graphs, Armendariz rings, idealization, Baer rings, nil-clean rings, annihilator ideals, quasi-Baer rings, factorization theory, triangular matrix representations, and semistar operations.

Keywords

Zero-Divisor Graphs; Armendariz Rings; Idealization; Baer Rings; Nil-Clean Rings; Annihilator Ideals; Quasi-Baer Rings; Factorization Theory; Triangular Matrix Representations; Semistar Operations

Linear Topological Spaces

1963-01-01

John L. Kelley , Isaac Namioka , William F. Donoghue +7 more

Finiteness Conditions and Generalized Soluble Groups

1972-01-01

Derek J. S. Robinson

Topics In Ring Theory

2010-01-01

I. N. Herstein

Modules over Non-Noetherian Domains

2000-11-14

L. Fuchs , Luigi Salce

Commutative domains and their modules Valuation domains Prufer domains More non-Noetherian domains Finitely generated modules Projectivity and projective dimension Divisible modules Topology and filtration Injective modules Uniserial modules Heights, invariants … Commutative domains and their modules Valuation domains Prufer domains More non-Noetherian domains Finitely generated modules Projectivity and projective dimension Divisible modules Topology and filtration Injective modules Uniserial modules Heights, invariants and basic submodules Polyserial modules RD- and pure-injectivity Torsion modules Torsion-free modules of finite rank Infinite rank torsion-free modules Appendix on set theory Bibliography Author index Subject index.

Rings and Categories of Modules

1974-01-01

Frank W. Anderson , Kent R. Fuller

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Seminar on transformation groups

1960-12-01

A. Borel , Glen E. Bredon

Abelian varieties over finite fields

1969-01-01

William C. Waterhouse

contenir la présente mention de copyright. contenir la présente mention de copyright.

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Multiplicative ideal theory

1968-01-01

Robert Gilmer

Foundations of Module and Ring Theory

2018-05-11

Wisbauer Robert

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful … This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Rings of Continuous Functions

1960-01-01

Leonard Gillman , Meyer Jerison

The Zero-Divisor Graph of a Commutative Ring

1999-07-01

David F. Anderson , Philip S. Livingston

Purity and algebraic compactness for modules

1969-03-01

Robert B. Warfield

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Algebraic approximation of structures over complete local rings

1969-01-01

Michael Artin

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Projective modules over polynomial rings

1976-12-01

Daniel Quillen

Tilting in abelian categories and quasitilted algebras

1996-01-01

Dieter Happel , Idun Reiten , O. SmaløSverre

Coloring of commutative rings

1988-07-01

István Beck

Vector bundles and projective modules

1962-01-01

Richard G. Swan

Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring.For some time, … Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring.For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X.A number of examples of projective modules have been given using this correspondence.However, no rigorous treatment of the correspondence seems to have been given.I will give such a treatment here and then give some of the examples which may be constructed in this way.1. Preliminaries.Let K denote either the real numbers, complex numbers or quaternions.A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z) = p-1(x), the structure of a finite dimensional vector space over K.These objects are required to satisfy the following condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homomorphism.The fibers u x Kn of U x K" are X-vector spaces in the obvious way.Note that I do not require n to be a constant.The dimension of the fiber Fx may vary with x.However, this dimension is clearly locally constant and so will be constant if X is connected.A subbundle of t] is, by definition, a subset Ex <= E (Ç) such that Ex n Fx is a X-subspace of Fx for each x and such that Ex with the projection p | Ex and the X-structure on its fibers induced by that of E forms a X-vector bundle over X.A map of X-vector bundles /:{-»i; is defined to be a continuous map f:E(Ç)-+E(n) such that pf' = p and such that /| Fx(0 : Fx(0 -► Fx(n) is a Xhomomorphism.It is clear that the X-vector bundles over X and their maps form an additive category.A section s of {; over a subset A cz X is a continuous map s : A -+ £(£) such that ps(x) = x.It follows immediately from the definition of a vector bundle that for any xeX, there is a neighborhood U of x and sections sx, ...,s" of £

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Lectures on Rings and Modules.

1969-11-01

Lawrence S. Levy , Joachim Lambek

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original … This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

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Direct products of modules

1960-01-01

Stephen U. Chase

Introduction.It is a well-known and basic result of homological algebra that the direct product of an arbitrary family of injective modules over any ring is again injective [3, p. 8].Such … Introduction.It is a well-known and basic result of homological algebra that the direct product of an arbitrary family of injective modules over any ring is again injective [3, p. 8].Such is not the case for projective modules, as is evidenced, for example, by a result of Baer [7, p. 48] which states that the direct product of a countably infinite number of copies of the ring of rational integers is not a free abelian group.It is thus natural to ask for the precise ideal-theoretic conditions which are forced upon a ring by the requirement that its projective modules be preserved by direct products in the manner just described.In this paper we shall present a solution to this problem, as well as an answer to the corresponding question for flat modules.We then exhibit several applications of these results.First, we derive information concerning semi-hereditary rings which, when applied to integral domains, yields immediately characterizations of Prüfer rings due to Hattori [5] (see also [6]).The second application also concerns integral domains.Let us call a torsion module over an integral domain £ a UT-module if it is a direct summand of every £-module of which it is the torsion submodule.We prove that, if every £-module of bounded order is a UT-module, then £ must be a Dedekind ring (the converse is well-known; see [8, p. 334]).As a final application, we obtain a partial solution to the following question of Köthe [9]: For which rings £ is it true that every left £-module is a direct sum of cyclic modules?We prove that, if £ has the weaker property that every left £-module is a direct sum of finitely generated modules, then £ satisfies the minimum condition on left ideals and every indecomposable injective left £-module has finite length.This generalizes a result of Kaplansky and Cohen [4].Our investigations along these lines were motivated to some extent by the interesting observation of Bass [2] that left Noetherian rings are characterized by the property that their injective left modules are preserved by direct sums.We have, with his kind permission, included this in our paper, as we shall need it in the proof of another result.

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Relative Homological Algebra

2011-08-23

Edgar E. Enochs , Overtoun M. G. Jenda

This second volume deals with the relative homological algebra of complexes of modules and their applications. It is a concrete and easy introduction to the kind of homological algebra which … This second volume deals with the relative homological algebra of complexes of modules and their applications. It is a concrete and easy introduction to the kind of homological algebra which has been developed in the last 50 years. The book serves as a bridge between the traditional texts on homological algebra and more advanced topics such as triangulated and derived categories or model category structures. It addresses to readers who have had a course in classical homological algebra, as well as to researchers.

Introduction To Commutative Algebra

2018-03-09

Michael Atiyah

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin … * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

The Brauer group of a commutative ring

1960-01-01

Maurice Auslander , Oscar Goldman

Let r be a ring having center C and let A be a C-subalgebra of I\ Then T may be considered a two-sided A-module.It Let r be a ring having center C and let A be a C-subalgebra of I\ Then T may be considered a two-sided A-module.It

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Free Rings and Their Relations.

1973-05-01

David J. Fieldhouse , P. M. Cohn

Applications of the theory of Boolean rings to general topology

1937-01-01

M. H. Stone

The notations of the present paper are largely determined by the usages already established in R. Thus, for instance, in considering Boolean rings as abstract entities, we continue to use … The notations of the present paper are largely determined by the usages already established in R. Thus, for instance, in considering Boolean rings as abstract entities, we continue to use the symbols -, v , +, <, ' corresponding to the symbols ■, u , A, c, ', of the theory of classes.In the algebra of the closure operation, we write A~ in place of A for convenience in putting down the more complicated expressions such as A~'~'.The contents of the present paper may be summarized systematically under the following headings: Chapter I, Boolean spaces: §1, Introduction of topological concepts; §2, Relations between algebra and topology; §3, Universal Boolean rings and spaces.Chapter II, Maps in Boolean spaces: §1, The general theory of maps; §2, Construction of Boolean maps; §3, Relation between algebraic and other maps; §4, Applications to the theory of extensions; §5, Totally-disconnected and discrete spaces.Chapter III, «Stronger separation conditions: §1

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Prime ideal structure in commutative rings

1969-01-01

M. Hochster

Augustfunctor oî3/t\n if; also, we shall say that Fis a space-preserving functor from 38 to«".It turns out that Spec can be inverted on surprisingly large subcategories of ¡f.At this point, … Augustfunctor oî3/t\n if; also, we shall say that Fis a space-preserving functor from 38 to«".It turns out that Spec can be inverted on surprisingly large subcategories of ¡f.At this point, the results of §7 should be read as part of this introduction.In fact, after looking at §1, §2 and the results of §7, the reader can continue and finish the paper (except for the proof of Theorem 9 in §16), and then go back to the technical § §3-6 if he wishes.We note that §16 uses the functorial nature of our constructions to characterize the underlying spaces of preschemes and schemes.

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A first course in noncommutative rings

1992-10-01

Tsit Yuen Lam

by similar or dissimilar methodology now known or hereafter developed is forbidden. by similar or dissimilar methodology now known or hereafter developed is forbidden.

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Injective modules over Noetherian rings

1958-09-01

Eben Matlis

Introduction In this discussion every module over a ring R will be understood to be a left i2-module. R will always have a unit, and every module will be unitary. … Introduction In this discussion every module over a ring R will be understood to be a left i2-module. R will always have a unit, and every module will be unitary. The aim of this paper is to study the structure and properties of injective modules, particularly over Noetherian rings. B. Eckmann and A. Schopf have shown that if M is a module over any ring, then there exists a unique, minimal, injective module E(M) containing it. The module E(M) will be a major tool in our investigations, and we shall systematically exploit its properties. In § 1 we show that if a module M has a maximal, injective submodule C (as is the case for left-Noetherian rings), then C contains a carbon-copy of every injective submodule of M, and MjC has no injective submodules different from 0. Although C is unique up to an automorphism of My C does not in general contain every injective submodule of M, In fact, the sum of two injective submodules of a module is always injective if and only if the ring is left-heredita ry. In § 2 we show that for any ring R a module E is an indecomposable, injective module if and only if E = E(R\J)y where J is an irreducible, left ideal of R. We prove that if R is a left-Noetherian ring, then every injective Jϋ-module has a decomposition as a direct sum of indecomposable, injective submodules. Strong uniqueness assertions can be made concerning such decompositions over any ring. In § 3 we take R to be a commutative, Noetherian ring, and to be a prime ideal of R. We prove there is a one-to-one correspondence between the prime ideals of R and the indecomposable, injective Rmodules given by P**E(RjP). We examine the structure of the module E = E{RjP)y and show that if At is the annihilator in E of P then E = U At and -4ί+1/At is a finite dimensional vector space over the quotient field of R/P. The ring of iϋ-endomorphi sms of E is isomorphic in a natural way to Rp, the completion of the ring of quotients of R with respect to R-P. As an ^-module E is an injective envelope of RpjP, where is the maximal ideal of Rp. If is a maximal ideal of Ry then E is a countably generated β-module. Every indecomposable, injective i2-module is finitely generated if and only if R has the minimum condition on ideals. In § 4 we take R to be a commutative, Noetherian, complete, local ring, the maximal ideal of R and E = E{RjP). Then the eontravariant,

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Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

1977-06-01

David A. Buchsbaum , David Eisenbud

Lifting idempotents and exchange rings

1977-01-01

W. K. Nicholson

Idempotents can be lifted modulo a one-sided ideal <italic>L</italic> of a ring <italic>R</italic> if, given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> … Idempotents can be lifted modulo a one-sided ideal <italic>L</italic> of a ring <italic>R</italic> if, given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x minus x squared element-of upper L"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x - {x^2} \in L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists an idempotent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">e \in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e minus x element-of upper L"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">e - x \in L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module <italic>P</italic> has the finite exchange property if and only if, whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals upper N plus upper M"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">P = N + M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <italic>N</italic> and <italic>M</italic> are submodules, there is a decomposition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals upper A circled-plus upper B"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">P = A \oplus B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to upper N"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A \subseteq N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B subset-of-or-equal-to upper M"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B \subseteq M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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On Rings of Operators. Reduction Theory

1949-04-01

John von Neumann

On the Structure of Semigroups

1951-07-01

J. A. Green

The purpose of this paper is to give the basis, and a few fundamental theorems, of a suggested systematic theory of semigroups. By a semigroup is meant a set S … The purpose of this paper is to give the basis, and a few fundamental theorems, of a suggested systematic theory of semigroups. By a semigroup is meant a set S closed to a single associative, binary multiplication. Two elements of S are said to be left equivalent (I) if they generate the same left ideal in S. Similarly a right equivalence (r) can be defined. These equivalences commute (or are associable, Dubreil, 1), and their product equivalence is denoted by b. The equivalence b is to be compared with the two-sided analogue f of I and r; x 3 y(f) means that x and y generate the same two-sided ideal. In the case where S is finite, these equivalences b and f coincide, but this is not true in general. A b-class has the property that the 1-classes which it contains generate isomorphic left ideals in S; a theorem on the structure of b-classes is given (Theorem 1). Schwarz (1) and Clifford (1, 3) have both made use of minimal conditions on right and left ideals; and Clifford has shown (1) that the minimum ideal of a semigroup, if it contains both minimal right, and minimal left ideals, is a simple subsemigroup of the type called by Rees completely simple (Rees, 1, 2). In his paper On Semigroups (Rees, 1), Rees determined the structure of such simple semigroups. The minimal conditions we use are more stringent, they are the minimal conditions on the partially ordered sets of the right, left and two-sided ideals. If the right and left conditions are satisfied, then so is the twosided condition (a fact not obvious in our case, since a two-sided ideal is not in general a right or left ideal). Further, in this case we do have b = f; and the b-classes take on a simpler aspect. The f-classes of any semigroup S correspond to semigroups called factors (cf. Rees, 1, p. 391) of S; our definition does not, however, depend on the existence of a principal series of ideals of S. We say that S is semisimple if all these factors are non-nilpotent semigroups. Another, and probably more fruitful idea, is that of regularity. The element a e S is regular if aza = a for some z e S; this is the condition introduced, for rings, by J. v. Neumann (1).1 A semigroup is regular, if all its elements are regular. Regular semigroups are semisimple, but not conversely; however, any semisimple semigroup which satisfies the right and left minimal conditions, is regular. It seems that these last-mentioned semigroups may form the class which will most repay study; the simple ones are completely simple (and therefore of known structure), and already the extension theory of Clifford (2) suggests the possibility of building up more

Rings of Continuous Functions.

1961-05-01

E. R. Lorch , Leonard Gillman , Meyer Jerison

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On Rings of Operators

1936-01-01

F. J. Murray , J. v. Neumann

Rings of Quotients

1975-01-01

Bo Stenström

Generalized Metric Spaces

1984-01-01

Gary Gruenhage

The Integers and Topology

1984-01-01

Eric K. van Douwen

LINEAR ALGEBRAIC GROUPS

1970-11-01

B. A. F. Wehrfritz

By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969). By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).

Presentations of faithful d.g. near-rings

1980-02-01

J. D. P. Meldrum

Many basic definitions and results in the theory of near-rings can be found in G. Pilz ( 4 ). We follow these for the most part, except that we use … Many basic definitions and results in the theory of near-rings can be found in G. Pilz ( 4 ). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum ( 2 ), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S , which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by ( R, S ). Then ( R , +) = Gp < S ; > where is a set of defining relations in S . Let ( T, U ) be a d.g. near-ring. Then a d.g. homomorphism θ from ( R, S ) to ( T, U ) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U . If ( G , +) is a group, let T 0 ( G ) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G . Then ( E ( G ), End G ) is a d.g. near-ring. A d.g. near-ring ( R, S ) is faithful if there exists a d.g. monomorphism θ:( R, S ) → ( E ( G ), End G ) for some group G .

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Commutative Coherent Rings

1989-01-01

Sarah Glaz

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Graded Ring Theory

1982-01-01

Lectures on modules and rings

1999-10-01

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Structure of Rings

1956-12-31

N. Jacobson

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Rings and Categories of Modules

1992-01-01

Frank W. Anderson , Kent R. Fuller

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The Divisor Class Group of a Krull Domain

1973-01-01

Robert M. Fossum

Trivial ring extension with epi-ACC on ideals

2025-06-25

Mohamed Khalifa | Ricerche di Matematica

Zero Intersection Graph of Annihilator Ideals of Modules

2025-06-24

Seçil Çeken , Osama A. Naji , Ünsal Teki̇̀r +1 more | Hacettepe Journal of Mathematics and Statistics

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let R be a commutative ring having a nonzero identity and M be a nonzero … This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let R be a commutative ring having a nonzero identity and M be a nonzero unital R-module. The zero intersection graph of annihilator ideals of R-module M, denoted by CR(M), is a simple (undirected) graph whose vertex set M⋆ = M − {0}, and two distinct vertices m and m′ are adjacent if annR(m)∩annR(m′) = (0). We investigate the conditions under which CR(M) is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property T in terms of their graphical properties.

Tight submodules over valuation domains

2025-06-24

Peter Danchev , L. Fuchs | Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

The maximal ring of quotients of $$A_d L$$

2025-06-23

Warren Wm. McGovern , Batsile Tlharesakgosi | Algebra Universalis

Relative $r$-noncommuting graph of finite rings

2025-06-22

M. Sharma , R. Nath | Carpathian Mathematical Publications

Let $S$ be a subring of a finite ring $R$ and $r\in R$. The relative $r$-noncommuting graph of $R$ relative to $S$, denoted by $\Gamma_{S, R}^r$, is a simple undirected … Let $S$ be a subring of a finite ring $R$ and $r\in R$. The relative $r$-noncommuting graph of $R$ relative to $S$, denoted by $\Gamma_{S, R}^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $x \in S$ or $y \in S$ and $[x,y] \neq r$, $[x,y]\neq -r$. In this paper, we determine degree of any vertex in $\Gamma_{S, R}^r$ and characterize all finite rings such that $\Gamma_{S, R}^r$ is a star, lollipop or a regular graph. We derive connections between relative $r$-noncommuting graphs of two isoclinic pairs of rings. We also derive certain relations between the number of edges in $\Gamma_{S, R}^r$ and various generalized commuting probabilities of $R$. Finally, we conclude the paper by studying an induced subgraph of $\Gamma_{S, R}^r$.

A generalization of primary ideals of commutative rings

2025-06-20

Najib Mahdou , Sanae Moussaoui , Ünsal Teki̇̀r +1 more | Asian-European Journal of Mathematics

ON THE CENTRE OF CROSSED MODULES OF GROUPS

2025-06-20

Mariam Pirashvili | Journal of Mathematical Sciences

On the prime spectrum of the amalgamations

2025-06-20

Y. Azimi | Communications in Algebra

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Subinjectivity Relative to Cotorsion Pairs

2025-06-18

Yusuf Alagöz , Rafai̇l Ali̇zade , Engi̇n Büyükaşık +2 more | Mathematics

In this paper, we define and study the X-subinjectivity domain of a module M where X=(A,B) is a complete cotorsion pair, which consists of those modules N such that, for … In this paper, we define and study the X-subinjectivity domain of a module M where X=(A,B) is a complete cotorsion pair, which consists of those modules N such that, for every extension K of N with K/N in A, any homomorphism f:N→M can be extended to a homomorphism g:K→M. This approach allows us to characterize some classical rings in terms of these domains and generalize some known results. In particular, we classify the rings with X-indigent modules—that is, the modules whose X-subinjectivity domains are as small as possible—for the cotorsion pair X=(FC,FI), where FI is the class of FP-injective modules. Additionally, we determine the rings for which all (simple) right modules are either X-indigent or FP-injective. We further investigate X-indigent Abelian groups in the category of torsion Abelian groups for the well-known example of the flat cotorsion pair X=(FL,EC), where FL is the class of flat modules.

Exact Decompositions and Zero-Divisor Graphs

2025-06-18

Justin Hoffmeier | Graphs and Combinatorics

Skew power series rings over von Neumann regular rings and a question on left quasi-morphic rings

2025-06-17

S Soumya , Sharad Hegde | Communications in Algebra

Modulus on Planar Maps

2025-06-17

Nathan Albin , Pietro Poggi‐Corradini | Chapman and Hall/CRC eBooks

Modulus on Graphs

2025-06-17

Nathan Albin , Pietro Poggi‐Corradini | Chapman and Hall/CRC eBooks

Stokes structure of mild difference modules

2025-06-17

Yota SHAMOTO | Bulletin de la Société mathématique de France

Characterization and examples of commutative iso-Artinian rings

2025-06-17

Asghar Daneshvar , Kamran Divaani-Aazar | Publicacions Matemàtiques

$\sqrt{J}U$ rings

2025-06-17

Shiksha Saini , Dinesh Udar | Czechoslovak Mathematical Journal

Diversity and Semiconvergents in Pythagorean Tuning

2025-06-17

Rafael Cubarsí | Axioms

Several approaches to building generalized Pythagorean scales provide interpretation for the rational approximation of an irrational number. Generally, attention is paid to the convergents of the continued fraction expansions. The … Several approaches to building generalized Pythagorean scales provide interpretation for the rational approximation of an irrational number. Generally, attention is paid to the convergents of the continued fraction expansions. The present paper focuses on the sequences of semiconvergents corresponding to the alternating best one-sided approximations. These sequences are interpreted as scale lineages organized as a kinship. Their properties are studied in terms of the two types of tones and elementary intervals, since each scale contains the tones of the previous scale plus the newly added tones, i.e., the generic diatones and accidentals. For the last scale of a lineage, the octave is regularly subdivided by sections, separated by a single elementary interval of the other type. Lineages are therefore related to the scale diversity with regard to their generic diatones and accidentals, which is analyzed from the Shannon diversity index, either for tone abundance or interval occupancy.

On the sum of annihilators in nearrings

2025-06-17

Nastaran Naseripour , Ebrahim Hashemı , Mahsa Paykanian | Quaestiones Mathematicae

A graph associated with tri-potent elements of commutative ring R

2025-06-14

Shaymaa S. Essa , Patrick Ali | Gulf Journal of Mathematics

In this paper, we introduce the tri-potent graph of a commutative ring R, denoted by TP(R), where two distinct vertices x and y in R are adjacent if and only … In this paper, we introduce the tri-potent graph of a commutative ring R, denoted by TP(R), where two distinct vertices x and y in R are adjacent if and only if (x + y)3 = x + y. We conduct a comprehensive investigation of the graphical structural properties of tri-potent graph of a commutative ring R, including its diameter, connectedness, and size. It is shown that the tri-potent graph of a commutative ring R contains cycles with girth 3 and has no end vertices. Furthermore, we describe a significant spanning subgraph of the tri-potent graph of a commutative ring R and analyze the degree of each vertex in detail. Finally, we establish that, for a specific local ring, tri-potent graph of a commutative ring R forms a 3-partite graph, discuss its planarity, and determine its independence domination number.

Completely Centrally Essential Rings

2025-06-13

O. V. Lyubimtsev , A. A. Tuganbaev | International Journal of Algebra and Computation

The complete integral closure of a Prüfer domain is a topological property

2025-06-13

Dario Spirito | Rendiconti del Circolo Matematico di Palermo Series 2

On subinjectivity domains of finitely generated modules

2025-06-12

Yılmaz Durğun , Salahattin Özdemi̇r | Communications in Algebra

An extension of semicommutative rings via reflexivity

2025-06-11

Sanjiv Subba , Tikaram Subedi | Annals of the Alexandru Ioan Cuza University - Mathematics

This article introduces the notion of an NJ-reflexive ring and demonstrates that it is distinct from the concept of a reflexive ring. The class of NJ-reflexive rings contains the class … This article introduces the notion of an NJ-reflexive ring and demonstrates that it is distinct from the concept of a reflexive ring. The class of NJ-reflexive rings contains the class of semicommutative rings, the class of left (right) quasi-duo rings, and the class of J-clean rings but is strictly larger than these classes. Additionally, the article investigates a sufficient condition for NJ-reflexive rings to be left (right) quasi-duo, as well as some conditions for NJ-reflexive rings to be reduced. It also explores extensions of NJreflexive rings and notes that the NJ-reflexive property may not carry over to polynomial extensions.

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On singular moduli for higher rank Drinfeld modules

2025-06-11

Chien-Hua Chen | Research in the Mathematical Sciences

The structure of S -semiprime ideals and submodules in noncommutative rings

2025-06-06

Alaa Abouhalaka , N. J. Groenewald | Communications in Algebra

Direct Projective Modules and Its Generalizations

2025-06-06

Ashok Ji Gupta , Sonal Gupta | Journal of Algebra and Its Applications

Matsuda monoids and Artin's primitive root conjecture

2025-06-06

Sunil Naik | International Journal of Number Theory

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Projectivity and Quasi-Projectivity with Respect to Epimorphisms to Simple Modules

2025-06-03

Yusuf Alagöz , Rafail Alizade , Engı̇n Büyükaşık

Algebraic Properties of Semi-Direct Sums of Rings

2025-06-02

Marta Nowakowska | Annales Mathematicae Silesianae

Abstract Let R be an associative ring not necessarily with unity. We say that R is a semi-direct sum of rings S and I , if R = S + … Abstract Let R be an associative ring not necessarily with unity. We say that R is a semi-direct sum of rings S and I , if R = S + I , where S is a subring of a ring R , I is an ideal of R and S ∩ I = {0}. The aim of this paper is to investigate certain algebraic properties of semidirect sums of associative rings with applications to amalgamated rings. We generalize several results from the literature to associative rings without unity. In particular we show that the class of semi-direct sums of rings is equal to the class of amalgamated rings, we provide a description of the Jacobson radical of semi-direct sums and we offer a characterization of semi-direct sums that are left Steinitz rings.

Partially ordered rings with negative squares

2025-06-02

Nhi Durtschi , Jingjing Ma , R. H. Redfield | Quaestiones Mathematicae

Weakly right coherent monoids

2025-06-02

Levent Michael Dasar , Victoria Gould , D. Craig Miller

Multiplication of SO(3) irreps in endomorphism algebras

2025-06-01

Yury Grabovsky | IOP Publishing eBooks

The Bass–Quillen Conjecture for Torsors Over Valuation Rings

2025-06-01

Ning Guo , Fei Liu | International Mathematics Research Notices

Abstract For a valuation ring $V$, a smooth $V$-algebra $A$, and a reductive $V$-group scheme $G$ satisfying a certain natural isotropicity condition, we prove that every Nisnevich $G$-torsor on $\mathbb{A}^{N}_{A}$ … Abstract For a valuation ring $V$, a smooth $V$-algebra $A$, and a reductive $V$-group scheme $G$ satisfying a certain natural isotropicity condition, we prove that every Nisnevich $G$-torsor on $\mathbb{A}^{N}_{A}$ descends to a $G$-torsor on $A$. As a corollary, we generalize Raghunathan’s theorem on torsors over affine spaces to a relative setting. We also extend several affine representability results of Asok, Hoyois, and Wendt from equi-characteristics to mixed characteristics. Our proof relies on previous work on the purity of reductive torsors over smooth relative curves and the Grothendieck–Serre conjecture for constant reductive group schemes.

Monoform Modules Relative to a Pure Property

2025-05-30

Muna Abbas Ahmed , Dunya Abdalhamed Hasan | Iraqi Journal of Science

The monoform concept is defined as a module in which every nonzero submodule is rational. The primary goal of this research is to study this class of modules in terms … The monoform concept is defined as a module in which every nonzero submodule is rational. The primary goal of this research is to study this class of modules in terms of pure property, named purely monoform modules. It constitutes an extension of the monoform modules; in fact, the monoform modules are properly included in the class of purely monoform modules. Many characteristics of a purely monoform module have been offered as analogues to those in monoform modules. A discussion of how this class of modules relates to other related modules is considered, like almost monoform, purely uniform, purely quasi-Dedekind and purely prime modules. Besides that, other characterizations of the purely monoform module have been given similar to those known and satisfied in the monoform modules.

(α, , n)-Multipliers on ring and C-Algebra

2025-05-30

Asma Ali , Faiza Shujat , Tooba Naz +1 more

Functional degrees and arithmetic applications III: Beyond prime exponent

2025-05-30

Pete L. Clark , Uwe Schauz

Abstract Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p -adic divisibility of the cardinality of the set of simultaneous … Abstract Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p -adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$ .

On Graded-(m,n)-Prime Submodules of Graded Modules

2025-05-29

Anass Assarrar

Artinian Serial Rings S with 𝒥n(S) = 0

2025-05-29

Yasser Ibrahim , Ahmed Aboubakr