Flat Modules Over Valuation Rings Francois Couchot
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Pub . Mat . UAB Vol . 27 N- 1 on the ENDOMORPHISM RING of A
Pub . Mat . UAB Vol . 27 n- 1 ON THE ENDOMORPHISM RING OF A FREE MODULE Pere Menal Throughout, let R be an (associative) ring (with 1) . Let F be the free right R-module, over an infinite set C, with endomorphism ring H . In this note we first study those rings R such that H is left coh- erent .By comparison with Lenzing's characterization of those rings R such that H is right coherent [8, Satz 41, we obtain a large class of rings H which are right but not left coherent . Also we are concerned with the rings R such that H is either right (left) IF-ring or else right (left) self-FP-injective . In particular we pro- ve that H is right self-FP-injective if and only if R is quasi-Frobenius (QF) (this is an slight generalization of results of Faith and .Walker [31 which assure that R must be QF whenever H is right self-i .njective) moreover, this occurs if and only if H is a left IF-ring . On the other hand we shall see that if' R is .pseudo-Frobenius (PF), that is R is an . injective cogenera- tortin Mod-R, then H is left self-FP-injective . Hence any PF-ring, R, that is not QF is such that H is left but not right self FP-injective . A left R-module M is said to be FP-injective if every R-homomorphism N -. M, where N is a -finitély generated submodule of a free module F, may be extended to F . -
Projective, Flat and Multiplication Modules
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 31 (2002), 115-129 PROJECTIVE, FLAT AND MULTIPLICATION MODULES M a j i d M . A l i a n d D a v i d J. S m i t h (Received December 2001) Abstract. In this note all rings are commutative rings with identity and all modules are unital. We consider the behaviour of projective, flat and multipli cation modules under sums and tensor products. In particular, we prove that the tensor product of two multiplication modules is a multiplication module and under a certain condition the tensor product of a multiplication mod ule with a projective (resp. flat) module is a projective (resp. flat) module. W e investigate a theorem of P.F. Smith concerning the sum of multiplication modules and give a sufficient condition on a sum of a collection of modules to ensure that all these submodules are multiplication. W e then apply our result to give an alternative proof of a result of E l-B ast and Smith on external direct sums of multiplication modules. Introduction Let R be a commutative ring with identity and M a unital i?-module. Then M is called a multiplication module if every submodule N of M has the form IM for some ideal I of R [5]. Recall that an ideal A of R is called a multiplication ideal if for each ideal B of R with B C A there exists an ideal C of R such that B — AC. Thus multiplication ideals are multiplication modules. In particular, invertible ideals of R are multiplication modules. -
Modules Embedded in a Flat Module and Their Approximations
International Journal of Recent Development in Engineering and Technology Website: www.ijrdet.com (ISSN 2347 - 6435 (Online)) Volume 2, Issue 3, March 2014) Modules Embedded in a Flat Module and their Approximations Asma Zaffar1, Muhammad Rashid Kamal Ansari2 Department of Mathematics Sir Syed University of Engineering and Technology Karachi-75300S Abstract--An F- module is a module which is embedded in a Finally, the theory developed above is applied to flat flat module F. Modules embedded in a flat module approximations particularly the I (F)-flat approximations. demonstrate special generalized features. In some aspects We start with some necessary definitions. these modules resemble torsion free modules. This concept generalizes the concept of modules over IF rings and modules 1.1 Definition: Mmod-A is said to be a right F-module over rings whose injective hulls are flat. This study deals with if it is embedded in a flat module F ≠ M mod-A. Similar F-modules in a non-commutative scenario. A characterization definition holds for a left F-module. of F-modules in terms of torsion free modules is also given. Some results regarding the dual concept of homomorphic 1.2 Definition: A ring A is said to be an IF ring if its every images of flat modules are also obtained. Some possible injective module is flat [4]. applications of the theory developed above to I (F)-flat module approximation are also discussed. 1.3 Definition [Von Neumann]: A ring A is regular if, for each a A, and some another element x A we have axa Keyswords-- 16E, 13Dxx = a. -
FLAT MODULES OVER VALUATION RINGS 3 S1,...,Sn ∈ R(R) and Y1,...,Yn ∈ F Such That Φ(X) = S1y1 + ··· + Snyn
FLAT MODULES OVER VALUATION RINGS FRANC¸OIS COUCHOT Abstract. Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent. A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect. In this paper, we consider the following properties of modules: P-flatness, flat- ness, content flatness, local projectivity, finite projectivity and single projectivity. We investigate the relations between these properties when R is a PP-ring or a valu- ation ring. Garfinkel ([11]), Zimmermann-Huisgen ([22]), and Gruson and Raynaud ([13]) introduced the concepts of locally projective modules and strongly coherent rings and developed important theories on these. The notions of finitely projec- tive modules and π-coherent rings are due to Jones ([15]). -
Pure Exactness and Absolutely Flat Modules
Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 9, Number 2 (2014), pp. 123-127 © Research India Publications http://www.ripublication.com Pure Exactness and Absolutely Flat Modules Duraivel T Department of Mathematics, Pondicherry University, Puducherry, India, 605014, E-mail: [email protected] Mangayarcarassy S Department of Mathematics, Pondicherry Engineering College, Puducherry, 605014, India, E-mail: [email protected] Athimoolam R Department of Mathematics, Pondicherry Engineering College, Puducherry, 605014, India, E-mail: [email protected] Abstract We show that an R-module M is absolutely flat over R if and only if M is R- flat and IM is a pure submodule of M, for every finitely generated ideal I. We also prove that N is a pure submodule of M and M is absolutely flat over R, if and only if both N and M/N are absolutely flat over R. AMS Subject Classification: 13C11. Keywords: Absolutely Flat Modules, Pure Submodule and Pure exact sequence. 1. Introduction Throughout this article, R denotes a commutative ring with identity and all modules are unitary. For standard terminology, the references are [1] and [7]. The concept of absolutely flat ring is extended to modules as absolutely flat modules, and studied in [2]. An R module M is said to be absolutely flat over R, if for every 2 Duraivel T, Mangayarcarassy S and Athimoolam R R-module N, M ⊗R N is flat R-module. Various characterizations of absolutely flat modules are studied in [2] and [4]. In this article, we show that an R-module M is absolutely flat over R if and only if for every finitely generated ideal I, IM is a pure submodule of M and M is R- flat. -
Reflexivity Revisited
REFLEXIVITY REVISITED MOHSEN ASGHARZADEH ABSTRACT. We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or close to free modules. 1. INTRODUCTION In this note (R, m, k) is a commutative noetherian local ring and M is a finitely generated R- module, otherwise specializes. The notation stands for a general module. For simplicity, M the notation ∗ stands for HomR( , R). Then is called reflexive if the natural map ϕ : M M M M is bijection. Finitely generated projective modules are reflexive. In his seminal paper M→M∗∗ Kaplansky proved that projective modules (over local rings) are free. The local assumption is really important: there are a lot of interesting research papers (even books) on the freeness of projective modules over polynomial rings with coefficients from a field. In general, the class of reflexive modules is extremely big compared to the projective modules. As a generalization of Seshadri’s result, Serre observed over 2-dimensional regular local rings that finitely generated reflexive modules are free in 1958. This result has some applications: For instance, in the arithmetical property of Iwasawa algebras (see [45]). It seems freeness of reflexive modules is subtle even over very special rings. For example, in = k[X,Y] [29, Page 518] Lam says that the only obvious examples of reflexive modules over R : (X,Y)2 are the free modules Rn. Ramras posed the following: Problem 1.1. (See [19, Page 380]) When are finitely generated reflexive modules free? Over quasi-reduced rings, problem 1.1 was completely answered (see Proposition 4.22). -
Arxiv:2002.10139V1 [Math.AC]
SOME RESULTS ON PURE IDEALS AND TRACE IDEALS OF PROJECTIVE MODULES ABOLFAZL TARIZADEH Abstract. Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f)+I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results. 1. Introduction and Preliminaries The concept of the trace ideals of modules has been the subject of research by some mathematicians around late 50’s until late 70’s and has again been active in recent years (see, e.g. [3], [5], [7], [8], [9], [11], [18] and [19]). This paper deals with some results on the trace ideals of projective modules. We begin with a few results on pure ideals which are used in their comparison with trace ideals in the sequel. After a few preliminaries in the present section, in section 2 a new characterization of pure ideals is given (Theorem 2.1) which is followed by some corol- laries. Section 3 is devoted to the trace ideal of projective modules. Theorem 3.1 gives a characterization of the trace ideal of a projective module in terms of the ideal generated by the “coordinates” of the ele- ments of the module. This characterization enables us to deduce some new results on the trace ideal of projective modules like the statement arXiv:2002.10139v2 [math.AC] 13 Jul 2021 on the trace ideal of the tensor product of two modules for which one of them is projective (Corollary 3.6), and some alternative proofs for a few known results such as Corollary 3.5 which shows that the trace ideal of a projective module is a pure ideal. -
Commutative Algebra Example Sheet 2
Commutative Algebra Michaelmas 2010 Example Sheet 2 Recall that in this course all rings are commutative 1. Suppose that S is a m.c. subset of a ring A, and M is an A-module. Show that there is a natural isomorphism AS ⊗A M =∼ MS of AS-modules. Deduce that AS is a flat A-module. 2. Suppose that {Mi|i ∈ I} is a set of modules for a ring A. Show that M = Li∈I Mi is a flat A-module if and only if each Mi is a flat A-module. Deduce that projective modules are flat. 3. Show that an A-module is flat if every finitely generated submodule is flat. 4. Show that flatness is a local properties of modules. 5. Suppose that A is a ring and φ: An → Am is an isomorphism of A-modules. Must n = m? 6. Let A be a ring. Suppose that M is an A-module and f : M → M is an A-module map. (i) Show by considering the modules ker f n that if M is Noetherian as an A-module and f is surjective then f is an isomorphism (ii) Show that if M is Artinian as an A-module and f is injective then f is an isomorphism. 7. Let S be the subset of Z consisting of powers of some integer n ≥ 2. Show that the Z-module ZS/Z is Artinian but not Noetherian. 8. Show that C[X1, X2] has a C-subalgebra that is not a Noetherian ring. 9. Suppose that A is a Noetherian ring and A[[x]] is the ring of formal power series over A. -
TENSOR PRODUCTS Balanced Maps. Note. One Can Think of a Balanced
TENSOR PRODUCTS Balanced Maps. Note. One can think of a balanced map β : L × M → G as a multiplication taking its values in G. If instead of β(`; m) we write simply `m (a notation which is often undesirable) the rules simply state that this multiplication should be distributive and associative with respect to the scalar multiplication on the modules. I. e. (`1 + `2)m = `1m + `2m `(m1 + m2)=`m1 + `m2; and (`r)m = `(rm): There are a number of obvious examples of balanced maps. (1) The ring multiplication itself is a balanced map R × R → R. (2) The scalar multiplication on a left R-module is a balanced map R × M → M . (3) If K , M ,andN are left R-modules and E =EndRMthen for δ ∈ EndR M and ' ∈ HomR(M; P) the product ϕδ is defined in the obvious way. This gives HomR(M; P) a natural structure as a right E-module. Analogously, HomR(K; M)isaleftE-moduleinanobviousway.ThereisanE-balanced map β :HomR(M; P) × HomR(K; M) → HomR(K; M)givenby( ;')7→ '. (4) If M and P are left R-modules and E =EndRM, then, as above, HomR(M; P) is a right E-module. Furthermore, M is a left E-module in an obvious way. We then get an E-balanced map HomR(M; P) × M → P by ('; m) 7→'(m). (5) A (distributive) multiplication on an abelian group G is a Z-balanced map G × G → G. It is interesting to note that in examples (1) through (4), the target space for the balancedmapisinfactanR-module, rather than merely an abelian group. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Stability, Duality, 2-Generated Ideals and a Canonical Decomposition of Modules Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 106 (2001), P
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA BRUCE OLBERDING Stability, duality, 2-generated ideals and a canonical decomposition of modules Rendiconti del Seminario Matematico della Università di Padova, tome 106 (2001), p. 261-290 <http://www.numdam.org/item?id=RSMUP_2001__106__261_0> © Rendiconti del Seminario Matematico della Università di Padova, 2001, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Stability, Duality, 2-Generated Ideals and a Canonical Decomposition of Modules. BRUCE OLBERDING (*) ABSTRACT - We study the relationship between stable domains, divisorial domains, the 2-generator property, Warfield duality and a canonical decomposition of modules. We classify Warfield domains (integral domains for which every overring is a reflexive domain), totally divisorial domains (integral domains for which every overring is a divisorial domain) and the class of integral do- mains that possess a canonical decomposition of torsion-free modules. 1. Introduction. In his «Ubiquity» paper of 1963, Bass shows how a particular concept of algebraic geometry can be expressed in a surprising number of ways. With the unifying notion of a Gorenstein ring, he interprets this concept through algebraic geometry (locally free sheaves of differentials on a va- riety), homological algebra (finite injective dimension), module theory (decompositions and reflexivity) and ideal theory (divisoriality and the 2- generator property). -
Flat Modules We Recall Here Some Properties of Flat Modules As
Flat modules We recall here some properties of flat modules as exposed in Bourbaki, Alg´ebreCommutative, ch. 1. The aim of these pages is to expose the proofs of some of the characterizations of flat and faithfully flat modules given in Matsumura's book Commutative Algebra. This part is mainly devoted to the exposition of a proof of the following Theorem. Let A be a commutative ring with identity and E an A-module. The following conditions are equivalent α β (a) for any exact sequence of A-modules 0 ! M 0 / M / M 00 ! 0 , the sequence 0 1E ⊗α 1E ⊗β 00 0 ! E ⊗A M / E ⊗A M / E ⊗A M ! 0 is exact; ∼ (b) for any finitely generated ideal I of A, the sequence 0 ! E ⊗A I ! E ⊗A A = E is exact; 0 b1 1 0 x1 1 t Pr . r . r (c) If bx = i=1 bixi = 0 for some b = @ . A 2 A and x = @ . A 2 E , there exist a matrix br xr s t A 2 Mr×s(A) and y 2 E such that x = Ay and bA = 0. α β We recall that, for any A-module, N, and for any exact sequence 0 ! M 0 / M / M 00 ! 0 , one gets the exact sequence 0 1N ⊗α 1N ⊗β 00 N ⊗A M / N ⊗A M / N ⊗A M ! 0 : Moreover, if αM 0 is a direct summand in M (which is equivalent to the existence of a morphism πM ! M 0 0 1N ⊗α such that πα = 1M 0 ), then the morphism N ⊗A M / N ⊗A M is injective and the image of 1N ⊗ α is a direct summand in N ⊗A M.