Semidistributive Modules and Rings Mathematics and Its Applications
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Dimension Theory and Systems of Parameters
Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite. -
Varieties of Quasigroups Determined by Short Strictly Balanced Identities
Czechoslovak Mathematical Journal Jaroslav Ježek; Tomáš Kepka Varieties of quasigroups determined by short strictly balanced identities Czechoslovak Mathematical Journal, Vol. 29 (1979), No. 1, 84–96 Persistent URL: http://dml.cz/dmlcz/101580 Terms of use: © Institute of Mathematics AS CR, 1979 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 29 (104) 1979, Praha VARIETIES OF QUASIGROUPS DETERMINED BY SHORT STRICTLY BALANCED IDENTITIES JAROSLAV JEZEK and TOMAS KEPKA, Praha (Received March 11, 1977) In this paper we find all varieties of quasigroups determined by a set of strictly balanced identities of length ^ 6 and study their properties. There are eleven such varieties: the variety of all quasigroups, the variety of commutative quasigroups, the variety of groups, the variety of abelian groups and, moreover, seven varieties which have not been studied in much detail until now. In Section 1 we describe these varieties. A survey of some significant properties of arbitrary varieties is given in Section 2; in Sections 3, 4 and 5 we assign these properties to the eleven varieties mentioned above and in Section 6 we give a table summarizing the results. 1. STRICTLY BALANCED QUASIGROUP IDENTITIES OF LENGTH £ 6 Quasigroups are considered as universal algebras with three binary operations •, /, \ (the class of all quasigroups is thus a variety). -
Commutative Algebra, Lecture Notes
COMMUTATIVE ALGEBRA, LECTURE NOTES P. SOSNA Contents 1. Very brief introduction 2 2. Rings and Ideals 2 3. Modules 10 3.1. Tensor product of modules 15 3.2. Flatness 18 3.3. Algebras 21 4. Localisation 22 4.1. Local properties 25 5. Chain conditions, Noetherian and Artin rings 29 5.1. Noetherian rings and modules 31 5.2. Artin rings 36 6. Primary decomposition 38 6.1. Primary decompositions in Noetherian rings 42 6.2. Application to Artin rings 43 6.3. Some geometry 44 6.4. Associated primes 45 7. Ring extensions 49 7.1. More geometry 56 8. Dimension theory 57 8.1. Regular rings 66 9. Homological methods 68 9.1. Recollections 68 9.2. Global dimension 71 9.3. Regular sequences, global dimension and regular rings 76 10. Differentials 83 10.1. Construction and some properties 83 10.2. Connection to regularity 88 11. Appendix: Exercises 91 References 100 1 2 P. SOSNA 1. Very brief introduction These are notes for a lecture (14 weeks, 2×90 minutes per week) held at the University of Hamburg in the winter semester 2014/2015. The goal is to introduce and study some basic concepts from commutative algebra which are indispensable in, for instance, algebraic geometry. There are many references for the subject, some of them are in the bibliography. In Sections 2-8 I mostly closely follow [2], sometimes rearranging the order in which the results are presented, sometimes omitting results and sometimes giving statements which are missing in [2]. In Section 9 I mostly rely on [9], while most of the material in Section 10 closely follows [4]. -
Unique Factorization of Ideals in a Dedekind Domain
UNIQUE FACTORIZATION OF IDEALS IN A DEDEKIND DOMAIN XINYU LIU Abstract. In abstract algebra, a Dedekind domain is a Noetherian, inte- grally closed integral domain of Krull dimension 1. Parallel to the unique factorization of integers in Z, the ideals in a Dedekind domain can also be written uniquely as the product of prime ideals. Starting from the definitions of groups and rings, we introduce some basic theory in commutative algebra and present a proof for this theorem via Discrete Valuation Ring. First, we prove some intermediate results about the localization of a ring at its maximal ideals. Next, by the fact that the localization of a Dedekind domain at its maximal ideal is a Discrete Valuation Ring, we provide a simple proof for our main theorem. Contents 1. Basic Definitions of Rings 1 2. Localization 4 3. Integral Extension, Discrete Valuation Ring and Dedekind Domain 5 4. Unique Factorization of Ideals in a Dedekind Domain 7 Acknowledgements 12 References 12 1. Basic Definitions of Rings We start with some basic definitions of a ring, which is one of the most important structures in algebra. Definition 1.1. A ring R is a set along with two operations + and · (namely addition and multiplication) such that the following three properties are satisfied: (1) (R; +) is an abelian group. (2) (R; ·) is a monoid. (3) The distributive law: for all a; b; c 2 R, we have (a + b) · c = a · c + b · c, and a · (b + c) = a · b + a · c: Specifically, we use 0 to denote the identity element of the abelian group (R; +) and 1 to denote the identity element of the monoid (R; ·). -
On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals
Canad. Math. Bull. Vol. 14 (3), 1971 ON PRIME RINGS WITH ASCENDING CHAIN CONDITION ON ANNIHILATOR RIGHT IDEALS AND NONZERO INFECTIVE RIGHT IDEALS BY KWANGIL KOH AND A. C. MEWBORN If / is a right ideal of a ring R91 is said to be an annihilator right ideal provided that there is a subset S in R such that I={reR\sr = 0, VseS}. lis said to be injective if it is injective as a submodule of the right regular i£-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective. LEMMA 1. Let M and T be right R-modules such that M is injective and T has zero singular submodule [4] and no nonzero injective submodule. Then Hom# (M, T)={0}. Proof. Suppose fe Hom^ (M, T) such that /=£ o. Let Kbe the kernel off. Then K is a proper submodule of M and there exists me M such that f(m)^0. Let (K:m)={reR\mreK}. Since the singular submodule of T is zero and f(m)(K:m)={0} the right ideal (K:m) has zero intersection with some nonzero right ideal / in R. Then ra/#{0} and K n mJ={0}. Let mJ be the injective hull of m J. Since M is injective, mJ is a submodule of M. mJC\ K={0} since m J has nonzero intersection with each submodule which has nonzero intersection with mJ (See [4, p. -
STRUCTURE THEORY of FAITHFUL RINGS, III. IRREDUCIBLE RINGS Ri
STRUCTURE THEORY OF FAITHFUL RINGS, III. IRREDUCIBLE RINGS R. E. JOHNSON The first two papers of this series1 were primarily concerned with a closure operation on the lattice of right ideals of a ring and the resulting direct-sum representation of the ring in case the closure operation was atomic. These results generalize the classical structure theory of semisimple rings. The present paper studies the irreducible components encountered in the direct-sum representation of a ring in (F II). For semisimple rings, these components are primitive rings. Thus, primitive rings and also prime rings are special instances of the irreducible rings discussed in this paper. 1. Introduction. Let LT(R) and L¡(R) designate the lattices of r-ideals and /-ideals, respectively, of a ring R. If M is an (S, R)- module, LT(M) designates the lattice of i?-submodules of M, and similarly for L¡(M). For every lattice L, we let LA= {A\AEL, AÍ^B^O for every nonzero BEL). The elements of LA are referred to as the large elements of L. If M is an (S, i?)-module and A and B are subsets of M, then let AB-1={s\sE.S, sBCA} and B~lA = \r\rER, BrQA}. In particu- lar, if ï£tf then x_10(0x_1) is the right (left) annihilator of x in R(S). The set M*= {x\xEM, x-WEL^R)} is an (S, i?)-submodule of M called the right singular submodule. If we consider R as an (R, i?)-module, then RA is an ideal of R called the right singular ideal in [6], It is clear how Af* and RA are defined and named. -
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. -
D-Extending Modules
Hacettepe Journal of Hacet. J. Math. Stat. Volume 49 (3) (2020), 914 – 920 Mathematics & Statistics DOI : 10.15672/hujms.460241 Research Article D-extending modules Yılmaz Durğun Department of Mathematics, Çukurova University, Adana, Turkey Abstract A submodule N of a module M is called d-closed if M/N has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through non- singular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands (respectively, closed, pure). The structure of a ring over which d-closed submodules of every module are direct summand (respectively, closed, pure) is studied. Mathematics Subject Classification (2010). 16D10, 16D40 Keywords. D-extending modules, d-closed submodules, semiartinian modules 1. Introduction Throughout we shall assume that all rings are associative with identity and all modules are unitary right modules. Let R be a ring and let M be an R-module. A (proper) submodule N of M will be denoted by (N M) N ≤ M. By E(M), Rad(M) and Soc(M) we shall denote the injective hull, the Jacobson radical, and the socle of M as usual. For undefined notions used in the text, we refer the reader to [2,11]. A submodule N of a module M is essential (or large) in M, denoted N ¢M, if for every 0 ≠ K ≤ M, we have N ∩ K ≠ 0; and N is said to be closed in M if N has no proper essential extension in M. We also say in this case that N is a closed submodule. -
INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 3 Contents 1
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 3 RAVI VAKIL Contents 1. Where we are 1 2. Noetherian rings and the Hilbert basis theorem 2 3. Fundamental definitions: Zariski topology, irreducible, affine variety, dimension, component, etc. 4 (Before class started, I showed that (finite) Chomp is a first-player win, without showing what the winning strategy is.) If you’ve seen a lot of this before, try to solve: “Fun problem” 2. Suppose f1(x1,...,xn)=0,... , fr(x1,...,xn)=0isa system of r polynomial equations in n unknowns, with integral coefficients, and suppose this system has a finite number of complex solutions. Show that each solution is algebraic, i.e. if (x1,...,xn) is a solution, then xi ∈ Q for all i. 1. Where we are We now have seen, in gory detail, the correspondence between radical ideals in n k[x1,...,xn], and algebraic subsets of A (k). Inclusions are reversed; in particular, maximal proper ideals correspond to minimal non-empty subsets, i.e. points. A key part of this correspondence involves the Nullstellensatz. Explicitly, if X and Y are two algebraic sets corresponding to ideals IX and IY (so IX = I(X)andIY =I(Y),p and V (IX )=X,andV(IY)=Y), then I(X ∪ Y )=IX ∩IY,andI(X∩Y)= (IX,IY ). That “root” is necessary. Some of these links required the following theorem, which I promised you I would prove later: Theorem. Each algebraic set in An(k) is cut out by a finite number of equations. This leads us to our next topic: Date: September 16, 1999. -
Refiltering for Some Noetherian Rings
Refiltering for some noetherian rings Noncommutative Geometry and Rings Almer´ıa, September 3, 2002 J. Gomez-T´ orrecillas Universidad de Granada Refiltering for some noetherian rings – p.1/12 Use filtrations indexed by more general (more flexible) monoids than Filtration Associated Graded Ring Re-filtering: outline Generators and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Associated Graded Ring Re-filtering: outline Generators Filtration and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Re-filtering: outline Use filtrations indexed by more general (more flexible) monoids than New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Keep (or even improve) properties of Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring We will show how these ideas can be used to obtain the Cohen-Macaulay property w.r.t. -
Integral Closures of Ideals and Rings Irena Swanson
Integral closures of ideals and rings Irena Swanson ICTP, Trieste School on Local Rings and Local Study of Algebraic Varieties 31 May–4 June 2010 I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found in Huneke–Swanson [13], but the lectures contain also more recent material. Table of contents: Section 1: Integral closure of rings and ideals 1 Section 2: Integral closure of rings 8 Section 3: Valuation rings, Krull rings, and Rees valuations 13 Section 4: Rees algebras and integral closure 19 Section 5: Computation of integral closure 24 Bibliography 28 1 Integral closure of rings and ideals (How it arises, monomial ideals and algebras) Integral closure of a ring in an overring is a generalization of the notion of the algebraic closure of a field in an overfield: Definition 1.1 Let R be a ring and S an R-algebra containing R. An element x S is ∈ said to be integral over R if there exists an integer n and elements r1,...,rn in R such that n n 1 x + r1x − + + rn 1x + rn =0. ··· − This equation is called an equation of integral dependence of x over R (of degree n). The set of all elements of S that are integral over R is called the integral closure of R in S. -
Localization in Non-Commutative Noetherian Rings
Can. J. Math., Vol. XXVIII, No. 3, 1976, pp. 600-610 LOCALIZATION IN NON-COMMUTATIVE NOETHERIAN RINGS BRUNO J. MÛLLER 1.1 Introduction and summary. To construct a well behaved localization of a noetherian ring R at a semiprime ideal 5, it seems necessary to assume that the set ^ (S) of modulo S regular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ring Rs in addition: one calls such 5 classical. To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal. The present paper studies the structure of classical semiprime ideals: they are built in a unique way from clans (minimal families of prime ideals with classical intersection), and each prime ideal belongs to at most one clan. We are thus led to regard the quotient rings Rs at the clans 5* as the natural local izations of a noetherian ring R. We determine these clans for rings which are finite as module over their centre, with an application to group rings, and for fflVP-rings, and provide some preliminary results for enveloping algebras of solvable Lie algebras. 1.2. Terminology. By a ring, we mean a not necessarily commutative ring with identity; and unless stated otherwise, a module is a unitary right-module. Terms like noetherian, ideal, etc. mean left- and right-noetherian, -ideal, etc. unless specified by one of the prefixes left- or right-.