Some modules with nearly prescribed endomorphism rings

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ20-hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π20-hard within the index set of computable rings. Finally, based on the Π20 definition of local rings, computable semiperfect rings can be described by Σ30 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ20-hard and Π20-hard within the index set of computable rings.

One-dimensional stable rings

The representation theory of curve singularities (more precisely, of their local rings) has turned out to be closely related to their deformation properties. Namely, as was shown in [6],[9],[7], such a ring R is of finite type, that is has only finitely many torsion-free indecomposable modules (up to isomorphism), if and only if it dominates one of the so called simple plane curve singularities in the sense of [1]. In [4] the authors have shown that R is of tame type, that is it has essentially only 1-parameter families of indecomposable torsion-free modules, if and only if it dominates one of the unimodal plane curve singularities of type T pq (T pq2 in the classification of [1]).

Given a field K K and a monoid M M with right cancelation we will construct an extension field F F such that the endomorphism monoid of F F is End ⁡ F = M ⋉ Frob ⁡ ( F ) , \operatorname {End} F= M\ltimes \operatorname {Frob}(F), where Frob ⁡ ( F ) \operatorname {Frob}(F) are the Frobenius endomorphisms of F F . Our main goal is to show that the equality, in particular Aut ⁡ F = G for groups G \operatorname {Aut} F=G \text { for groups } G holds absolutely . Thus the realization of M M by F F should hold in any generic extension of the universe, e.g., if we change the model of set theory by forcing arguments. The construction must be very robust, so, for example, applications of stationary sets (as in Dugas and Göbel, 1987 and 1994) are not permitted. From Herden-Shelah (2009) it follows that our claim can only hold if the cardinality of F F is less than the first ω \omega -Erdős cardinal. Like measurable cardinals, this is a huge cardinal. The formula for endomorphism monoids requires a mild restriction on K K (which is also necessary for Theorem 2.4), while for automorphism groups K K can be arbitrary. We will proceed as follows: An old paper (from 1982) by Shelah provides the existence of absolutely rigid trees. They can be encoded into P 4 P_4 -vector spaces (which are vector spaces V V with 4 4 distinguished subspaces) over the prime field P P of K K (thus P ∈ { Q , Z p } P\in \{\mathbb {Q}, Z_p\} ) such that the monoid P P -algebra A = P ( M ) A=P(M) can be represented absolutely as the endomorphism algebra A = End P ⁡ ( V , V 1 , … , V 4 ) A= \operatorname {End}_P(V, V_1,\dots , V_4) . Such a P 4 P_4 -vector space again will be encoded into a field F F which is close to a function field and is as desired. The existence of P 4 P_4 -vector spaces follows by combining earlier results Göbel and Shelah (2007), Fuchs and Göbel (2008), and Göbel (2011). We also extend arguments and results from two other earlier, non-absolute constructions, Dugas and Góbel (1994 and 1997).

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

Applications of dualizing complexes to Buchsbaum rings

Let N be a normal subgroup of a finite group G and let R be a noetherian complete local commutative ring. Clifford theory deals with the relationship between RG-modules and RN-modules, using induction from N to G or restriction from G to N. Since Clifford's 1937 paper [1], the theory is well understood for irreducible representations (see also [2, w For an indecomposable RNmodule W, several authors have proved a going-up theorem describing how Ind~ W decomposes (see [2, w One purpose of this paper is to prove (in Section 2) a going-down theorem for indecomposable modules (analogous to Clifford's theorem), based on a refinement of the lifting idempotents theorem, presented in Section 1. The going-up and going-down theorems are actually equivalent in the sense that each can be dcrivcd as a corollary to the other one. One main assumption is necessary for the going-down theorem: the RG-module we start from must be projective relative to H. The whole procedure is presented in the more general context of Clifford systems. The paper concludes in Section 3 with another application of the lifting idempotents theorem, concerning the behaviour of indecomposable modules under ground ring extensions.

It is well known that two basic rings are Morita equivalent if and only if they are isomorphic.Here it is shown that local uniserial rings are stably equivalent in the sense of M. Auslander and I. Reiten in case they are isomorphic modulo certain powers of their radicals.In particular, two commutative local uniserial rings of Loewy length n are stably equivalent if and only if they are isomorphic modulo the [/i/2]th power of their radicals.In a series of papers [2]-[6], M. Auslander and I. Reiten have studied categories of modules modulo projectives and the related notion of stable equivalence of rings.We recall their terminology.For a ring A, mod A denotes the category of finitely generated left A-modules.For M, N G mod A, PA(M, N) is the subgroup of HomA(M, N) consisting of the homomorphisms /: M -> N that factor through projectives.The additive category of finitely generated left A-modules modulo projectives is mod A. Its objects are the same as those of mod A, and its morphisms are the members of the factor groups HomA(A/, N) = HomA(A/, N)/PA(M, N).The category modF A is the full subcategory of mod A whose objects have no projective direct summands, and mod,, A is the corresponding subcategory of mod A. If A is a left artinian ring then mod,, A mod A ( denotes equivalence of categories) [3, p. 247].Two rings A and A' are said to be (left) stably equivalent in case mod A mod A'.Stable equivalence is a relaxing of the notion of Morita equivalence.So since artinian rings are Morita equivalent iff their basic rings are isomorphic (see [1, 27]), one is led to study the relative structure of stably equivalent rings.For example, although Zp2 and Zp[x]/(x2) are not Morita equivalent, they are stably equivalent [2].See also [6] and [7] regarding stable equivalence of serial rings.Here we consider stable equivalence of uniserial rings.A uniserial ring is one that is Morita equivalent to a direct sum of local uniserial rings.A ring A with J = J (A) and Loewy length t? is a local uniserial ring in case A > J > > J"~x > J" = 0 is a. complete list of the submodules of both AA and AA.The indecomposable modules over A are then A/J, ..., A/J"~x, A, and PA(A/J', A/Jj) consists of those maps that factor through A. In what

4 - Indecomposable Modules

No AccessTwisted Hessian curves over the ring 𝔽q[∈], ∈2 = 0Abdelâli Grini, Abdelhakim Chillali, Lhoussain El Fadil and Hakima MouanisAbdelâli GriniDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco, Abdelhakim ChillaliDepartment of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco, Lhoussain El FadilDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco and Hakima MouanisDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPublished Online:November 28, 2022pp 181-189https://doi.org/10.1504/IJCAET.2023.127795PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractThe goal of this work is to study some arithmetic proprieties of the twisted Hessian curves defined by a equation of type: aX3 + Y3 + Z3 = dXYZ on the local ring R2 = 𝔽q[X]/(X2), where p ≥ 5 is a prime number, q = pd and d ∈ ℕ*, such that -3 is not a square in 𝔽q. This paper consists of an introduction, section, and a conclusion. In the introduction, we review some fundamental arithmetic proprieties of finite local rings R2, which will be used in the remainder of this article. The section is devoted to the study the above mentioned twisted Hessian curves on these finite local rings for restriction to some specific characteristic p ≥ 5. Using these studies, we give essential properties and we define the group H2a,d, these properties, the classification of these elements and a bijection between the sets H2a,d and Ha0,d0 × 𝔽q, where Ha0,d0 is the twisted Hessian curve over the finite field 𝔽q.Keywordstwisted Hessian curves, finite over ring, cryptography, elliptic curves, local ring Previous article Next article FiguresReferencesRelatedDetails Volume 18Issue 1-32023 ISSN: 1757-2657eISSN: 1757-2665 HistoryPublished onlineNovember 28, 2022 Copyright © 2023 Inderscience Enterprises Ltd.Keywordstwisted Hessian curvesfinite over ringcryptographyelliptic curveslocal ringAuthors and AffiliationsAbdelâli Grini1 Abdelhakim Chillali2 Lhoussain El Fadil3 Hakima Mouanis4 1. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco2. Department of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco3. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco4. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPDF download

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.