A NOTE on MONOFORM MODULES 1. Introduction
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Injective Modules: Preparatory Material for the Snowbird Summer School on Commutative Algebra
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them. Let R be a commutative Noetherian ring with an identity element. An R- module E is injective if HomR( ;E) is an exact functor. The main messages of these notes are − Every R-module M has an injective hull or injective envelope, de- • noted by ER(M), which is an injective module containing M, and has the property that any injective module containing M contains an isomorphic copy of ER(M). A nonzero injective module is indecomposable if it is not the direct • sum of nonzero injective modules. Every injective R-module is a direct sum of indecomposable injective R-modules. Indecomposable injective R-modules are in bijective correspondence • with the prime ideals of R; in fact every indecomposable injective R-module is isomorphic to an injective hull ER(R=p), for some prime ideal p of R. The number of isomorphic copies of ER(R=p) occurring in any direct • sum decomposition of a given injective module into indecomposable injectives is independent of the decomposition. Let (R; m) be a complete local ring and E = ER(R=m) be the injec- • tive hull of the residue field of R. The functor ( )_ = HomR( ;E) has the following properties, known as Matlis duality− : − (1) If M is an R-module which is Noetherian or Artinian, then M __ ∼= M. -
SS-Injective Modules and Rings Arxiv:1607.07924V1 [Math.RA] 27
SS-Injective Modules and Rings Adel Salim Tayyah Department of Mathematics, College of Computer Science and Information Technology, Al-Qadisiyah University, Al-Qadisiyah, Iraq Email: [email protected] Akeel Ramadan Mehdi Department of Mathematics, College of Education, Al-Qadisiyah University, P. O. Box 88, Al-Qadisiyah, Iraq Email: [email protected] March 19, 2018 Abstract We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R- injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provide some new characterizations of universally mininjective rings, quasi-Frobenius rings, Artinian rings and semisimple rings. Key words and phrases: Small injective rings (modules); soc-injective rings (modules); SS- Injective rings (modules); Perfect rings; quasi-Frobenius rings. 2010 Mathematics Subject Classification: Primary: 16D50, 16D60, 16D80 ; Secondary: 16P20, 16P40, 16L60 . arXiv:1607.07924v1 [math.RA] 27 Jul 2016 ∗ The results of this paper will be part of a MSc thesis of the first author, under the supervision of the second author at the University of Al-Qadisiyah. 1 Introduction Throughout this paper, R is an associative ring with identity, and all modules are unitary R- modules. -
L-Stability in Rings and Left Quasi-Duo Rings
University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2018-05-28 L-stability in Rings and Left Quasi-duo Rings Horoub, Ayman Mohammad Abedalqader Horoub, A. M. A, (2018). L-stability in Rings and Left Quasi-duo Rings (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31958 http://hdl.handle.net/1880/106702 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY L-stability in Rings and Left Quasi-duo Rings by Ayman Mohammad Abedalqader Horoub A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS CALGARY, ALBERTA May, 2018 c Ayman Mohammad Abedalqader Horoub 2018 Abstract A ring R is said to have stable range 1 if, for any element a 2 R and any left ideal L of R, Ra + L = R implies a − u 2 L for some unit u in R. Here we insist only that this holds for all L in some non-empty set L(R) of left ideals of R, and say that R is left L-stable in this case. We say that a class C of rings is affordable if C is the class of left L-stable rings for some left idealtor L. -
On Continuous Rings
JOURNAL OF ALGEBRA 191, 495]509Ž. 1997 ARTICLE NO. JA966936 On Continuous Rings Mohamed F. Yousif* Department of Mathematics, The Ohio State Uni¨ersity, Lima, Ohio 45804 and Centre de Recerca Matematica, Institut d'Estudis Catalans, Apartat 50, E-08193 Bellaterra, Spain View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Kent R. Fuller provided by Elsevier - Publisher Connector Received March 11, 1996 We show that if R is a semiperfect ring with essential left socle and rlŽ. K s K for every small right ideal K of R, then R is right continuous. Accordingly some well-known classes of rings, such as dual rings and rings all of whose cyclic right R-modules are essentially embedded in projectives, are shown to be continuous. We also prove that a ring R has a perfect duality if and only if the dual of every simple right R-module is simple and R [ R is a left and right CS-module. In Sect. 2 of the paper we provide a characterization for semiperfect right self-injective rings in terms of the CS-condition. Q 1997 Academic Press According to S. K. Jain and S. Lopez-Permouth wx 15 , a ring R is called a right CEP-ring if every cyclic right R-module is essentially embedded in a projectiveŽ. free right R-module. In a recent and interesting article by J. L. Gomez Pardo and P. A. Guil Asensiowx 9 , right CEP-rings were shown to be right artinian. In this paper we will show that such rings are right continuous, and so R is quasi-Frobenius if and only if MR2Ž.is a right CEP-ring. -
Lectures on Local Cohomology
Contemporary Mathematics Lectures on Local Cohomology Craig Huneke and Appendix 1 by Amelia Taylor Abstract. This article is based on five lectures the author gave during the summer school, In- teractions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concern- ing the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals. Contents 1. Introduction 1 2. Local Cohomology 3 3. Injective Modules over Noetherian Rings and Matlis Duality 10 4. Cohen-Macaulay and Gorenstein rings 16 d 5. Vanishing Theorems and the Structure of Hm(R) 22 6. Vanishing Theorems II 26 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 32 8. Appendix 2: Bass numbers and Gorenstein Rings 37 References 41 1. Introduction Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization 2000 Mathematics Subject Classification. Primary 13C11, 13D45, 13H10. Key words and phrases. local cohomology, Gorenstein ring, initial ideal. The first author was supported in part by a grant from the National Science Foundation, DMS-0244405. -
Injective Hulls of Simple Modules Over Some Noetherian Rings
Osman Can Hatipoglu˘ Injective Hulls of Simple Modules Over Some Noetherian Rings Faculdade de Ciências da Universidade do Porto Osman Can Hatipoglu˘ Injective Hulls of Simple Modules Over Some Noetherian Rings Tese submetida à Faculdade de Ciências da Universidade do Porto para obtenção do grau de Doutor em Matemática Outubro de 2013 Resumo Estudamos a seguinte propriedade de finitude de um anel Noetheriano esquerdo R: (⋄) Os envolucros injectivos de R-módulos simples são localmente Artinianos. A propriedade (⋄) é estudada em duas classes de anéis. Motivados pelo resul- tado de Musson: nenhuma álgebra envolvente duma álgebra de Lie de dimensão finita solúvel mas não nilpotent sobre um corpo algebricamente fechado satisfaz a propriedade (⋄), começamos por considerar as super álgebras de Lie nilpotentes e descrevemos as de dimensão finita cuja álgebra envolvente satisfaz a propriedade (⋄). A segunda classe de anéis que estudamos são os anéis de operadores diferen- ciais sobre um anel de polinómios com coeficientes num corpo, S = k[x][y;δ]. Para esta classe de anéis obtemos condições suficientes para a existência de extensões essenciais de módulos simples que não são Artinianos. Combinando os obtidos com resultados de Awami, Van den Bergh, e van Oystaeyen e de Alev e Dumas relativa- mente a classificação das extensões de Ore obtemos a caracterização completa de extensões de Ore S = k[x][y;σ,δ] que satisfazem a propriedade (⋄). Consideramos ainda a relaço entre teorias de torção estáveis e a propriedade (⋄) em anéis Noetherianos. Em particular, mostramos que um anel Noetheriano R tem a propriedade (⋄) se e só se a teoria de torção de Dickson em R-Mod é estável. -
The Differential Graded Stable Category of a Self-Injective Algebra
The Differential Graded Stable Category of a Self-Injective Algebra Jeremy R. B. Brightbill July 17, 2019 Abstract Let A be a finite-dimensional, self-injective algebra, graded in non- positive degree. We define A -dgstab, the differential graded stable category of A, to be the quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab /Ω(1). This result allows computations in the dg-stable category to be performed by reducing to the graded stable category. We provide a sufficient condition for the orbit cat- egory to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed de- scription of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges. 1 Introduction If A is a self-injective k-algebra, then A -stab, the stable module category of A, arXiv:1811.08992v2 [math.RT] 18 Jul 2019 admits the structure of a triangulated category. This category has two equivalent descriptions. The original description is as an additive quotient: One begins with the category of A-modules and sets all morphisms factoring through projective modules to zero. More categorically, we define A -stab to be the quotient of ad- ditive categories A -mod /A -proj. The second description, due to Rickard [13], describes A -stab as a quotient of triangulated categories. Rickard obtains A -stab as the quotient of the bounded derived category of A by the thick subcategory of perfect complexes (i.e., complexes quasi-isomorphic to a bounded complex of projective modules). -
Study of Formal Triangular Matrix Rings
International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 ISSN 2229-5518 21 STUDY OF FORMAL TRIANGULAR MATRIX RINGS MRS Sunita Salunke SCOE Kharghar Navi Mumbai Email:[email protected] Abstract—There has been a continuous study on formal triangular matrix rings. In this present study 2. Generalities on formal triangular matrix of formal triangular matrix rings few ring theoretic rings characteristic properties of formal triangular matrix rings have been studied in detail. Some definitive re- Proposition II.1. (1) The set of maximal right sults are verified on these rings concerning properties ideals of T is given by such as being respectively left Kasch, right mininjec- tive, clean, potent or a ring of stable rank≤ n. The f(I ⊕ M) ⊕ Kj either I = concepts of a strong left Kasch ring and a strong right A and K is a maximal right ideal of B mininjective ring are introduced and it is determined or I is a maximal right ideal of A and K = when the triangular matrix rings are respectively Bg strong left Kasch or strong right mininjective (2) The set of minimal right ideals of T is the I. INTRODUCTION union of the two sets, fW ⊕ 0j W a simple submodule of (A ⊕ All the rings considered will be associative rings M) g and with identity, all the modules considered will be A f0 ⊕ Kj with 0 the zero submodule of (A ⊕ unital modules. For any ring R the category of right M) and K a minimal right (resp left) R-modules will be denoted by Mod- A ideal of B satisfying KM = 0g: R(resp R-Mod). -
Continuous Homomorphisms and Rings of Injective Dimension One
MATH. SCAND. 110 (2012), 181–197 CONTINUOUS HOMOMORPHISMS AND RINGS OF INJECTIVE DIMENSION ONE SHOU-TE CHANG and I-CHIAU HUANG Abstract Let S be an R-algebra and ᑾ be an ideal of S. We define the continuous hom functor from R-Mod to S-Mod with respect to the ᑾ-adic topology on S. We show that the continuous hom functor preserves injective modules iff the ideal-adic property and ideal-continuity property are satisfied for S and ᑾ. Furthermore, if S is ᑾ-finite over R, we show that the continuous hom functor also preserves essential extensions. Hence, the continuous hom functor can be used to construct injective modules and injective hulls over S using what we know about R. Using the continuous hom functor we can characterize rings of injective dimension one using symmetry for a special class of formal power series subrings. In the Noetherian case, this enables us to construct one- dimensional local Gorenstein domains. In the non-Noetherian case, we can apply the continuous hom functor to a generalized form of the D+M construction. We may construct a class of domains of injective dimension one and a series of almost maximal valuation rings of any complete DVR. 1. Introduction Throughout this paper, R, S and T are rings, S is an R-algebra and ᑾ is an ideal of S. When R is a domain we always use Q to denote Q(R), the field of fractions of R, and K to denote a field containing Q(R). When X is used it always stands for an indeterminate over whatever base ring used. -
T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules
Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 DOI: http://dx.doi.org/10.21123/bsj.2021.18.1.0156 T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules Inaam Mohammed Ali Hadi*1 Shukur Neamah Al-Aeashi 2 Farhan Dakhil Shyaa3 1 Department of Mathematics, College of Education for Pure Science/ Ibn Al-Haitham, University of Baghdad, Iraq. 2 Department of Mathematics, College of Education, University of Kufa, Kufa, Iraq. 3 Department of Mathematics, College of Education, University of Al-Qadisiyah, Qadisiyah, Iraq. *Corresponding author: [email protected] *,[email protected], [email protected] *ORCID ID: https://orcid.org/0000-0002-1058-7410, https://orcid.org/0000-0002-9886-3247, https://orcid.org/0000- 0003-1123-1974 Received 24/2/2019, Accepted 25/4/2020, Published Online First 6/12/2020, Published 1/3/2021 This work is licensed under a Creative Commons Attribution 4.0 International License. Abstract: A new generalizations of coretractable modules are introduced where a module ℳ is called t-essentially (weakly t-essentially) coretractable if for all proper submodule 퐾 of ℳ, there exists f∈End(ℳ), f(퐾)=0 and Imf≤tes ℳ (Im f +퐾 ≤tes ℳ). Some basic properties are studied and many relationships between these classes and other related one are presented. Key words: Coretractable module, Essentially coretractable module, T-essentially coretractable module, Weakly t-essentially coretractable module, Weakly essentially coretractable module. Introduction: In this work, all rings have identity and all weakly essentially coretractable if Imf+퐾 ≤ess ℳ" modules are unitary left R-modules. -
Divisible Is Injective
SOOCHOW JOURNAL OF MATHEMATICS Volume 32, No. 2, pp. 241-243, April 2006 DIVISIBLE IS INJECTIVE BY SHALOM FEIGELSTOCK Abstract. It is well known that injective modules are divisible. A result attributed to Cartan and Eilenberg states that if R is a commutative integral domain then a torsion free R-module is injective if and only if it is divisible. In this note the same result is obtained for a larger class of rings, namely rings R which embed in a ring S in which every element of R is invertible. R is a ring with unity, and is a subring of a ring S: All modules are unital left R-modules. The injective hull of a module M will be denoted by I(M). Definition. An R-module M is R-torsion free if rm = 0 for r 2 R and m 2 M, then either r = 0 or m = 0: Definition. An R-module M is R-divisible if rM = M for all 0 =6 r 2 R. Lemma 1. If every non-zero element of R is invertible in S then S is an injective R-module. Proof. Let 0 =6 x 2 I(S): There exists 0 =6 r 2 R such that rx 2 S; so x = r−1(rx) 2 S: Corollary 2. If S has no zero-divisors and if S is an R-divisible R-module then S is an injective R-module. Proof. Let 0 =6 r 2 R. There exists s 2 S such that rs = 1: Therefore srs = s and so (sr − 1)s = 0: Since S has no zero-divisors sr = 1; and S is injective by Lemma 1. -
ON SEMIPERFECT F-INJECTIVE RINGS Truong Cong Quynh 1
International Electronic Journal of Algebra Volume 1 (2007) 18{29 ON SEMIPERFECT F-INJECTIVE RINGS Truong Cong Quynh Received: 12 June 2006; Revised: 26 October 2006 Communicated by W. K. Nicholson Abstract. A ring R is called right F-injective if every right R-homomorphism from a finitely generated right ideal of R to R extends to an endomorphism of R. R is called a right FSE-ring if R is a right F-injective semiperfect ring with essential right socle. The class of right FSE-rings is broader than that of right PF-rings. In this paper, we study and provide some characterizations of this class of rings. We prove that if R is left perfect, right F-injective, then R is QF if and only if R=S is left finitely cogenerated where S = Sr = Sl if and only if R is left semiartinian, Soc2(R) is left finitely generated. It is also proved that R is QF if and only if R is left perfect, mininjective and J2 = r(I) for a finite subset I of R. Some known results are obtained as corollaries. Mathematics Subject Classification (2000): 16D50, 16D70, 16D80 Keywords: F(P)-injective ring, mininjective ring, finitely continuous ring, min-CS, QF-ring, PF-ring, FSE-ring, uniform module 1. Introduction Throughout the paper, R represents an associative ring with identity 1 6= 0 and all modules are unitary R-modules. We write MR (resp., RM) to indicate that M is a right (resp., left) R-module. We also write J (resp., Zr, Sr) for the Jacobson radical (resp., the right singular ideal, the right socle of R) and E(MR) for the injective hull of MR.