Perinormal Rings with Zero Divisors
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Polynomial Rings
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, No. 1, January 1972 A NOTE ON FINITE DIMENSIONAL SUBPJNGS OF POLYNOMIAL RINGS PAUL EAKIN Abstract. Let k be a field and {Xx}xea a family of indeterminates over k. We show that if A is a ring of Knill dimension d such that k c A c. k[{Ax}xe^] then there are elements Yt,''■' •, Y* which are algebraically independent over k and a fc-isomorphism <f>such that k c. (f>(A)c k[Ylt- ■■ , Yd]. This is used to show that a one- dimensional ring A which satisfies the above conditions is necessarily an affine ring over k and is necessarily a polynomial ring if it is nor- mal. In addition we show that such a ring A is a normal affine ring of transcendence degree two over k if and only if it is a two-dimensional Krull ring such that each essential valuation of A has residue field transcendental over k. Introduction. In [EZ] Evyatar and Zaks ask whether a Dedekind domain which is caught between a field k and the polynomials in a finite number of variables over k is necessarily a polynomial ring. Here we show that the answer is affirmative with no restriction on the number of variables and without assuming the ring satisfies all three of the Noether axioms on a Dedekind domain (it need only be one dimensional and integrally closed). In addition we give some general results which relate subrings of the polynomials over a field to noetherian rings. The author is indebted to W. -
The Center of Topologically Primitive Exponentially Galbed Algebras
THE CENTER OF TOPOLOGICALLY PRIMITIVE EXPONENTIALLY GALBED ALGEBRAS MART ABEL AND MATI ABEL Received 29 September 2004; Revised 11 December 2005; Accepted 18 December 2005 Let A be a unital sequentially complete topologically primitive exponentially galbed Haus- dorff algebra over C, in which all elements are bounded. It is shown that the center of A is topologically isomorphic to C. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction (1) Let A be an associative topological algebra over the field of complex numbers C with separately continuous multiplication. Then A is an exponentially galbed algebra (see, e.g., [1–4, 19, 20]) if every neighbourhood O of zero in A defines another neighbourhood U of zero such that n a k a ... a ∈ U ⊂ O k : 0, , n (1.1) k=0 2 for each n ∈ N. Herewith, A is locally pseudoconvex,ifithasabase{Uλ : λ ∈ Λ} of neigh- bourhoods of zero consisting of balanced and pseudoconvex sets (i.e., of sets U for which μU ⊂ U,whenever|μ| 1, and U + U ⊂ ρU for a ρ 2). In particular, when every Uλ in {Uλ : λ ∈ Λ} is idempotent (i.e., UλUλ ⊂ Uλ), then A is called a locally m-pseudoconvex algebra, and when every Uλ in {Uλ : λ ∈ Λ} is A-pseudoconvex (i.e., for any a ∈ A there is a μ>0suchthataUλ,Uλa ⊂ μUλ), then A is called a locally A-pseudoconvex algebra.Itiswell known (see [21,page4]or[6, page 189]) that the locally pseudoconvex topology on A is given by a family {pλ : λ ∈ Λ} of kλ-homogeneous seminorms, where kλ ∈ (0,1] for each λ ∈ Λ.Thetopologyofalocallym-pseudoconvex (A-pseudoconvex) algebra -
Derivations and Integral Closure
Pacific Journal of Mathematics DERIVATIONS AND INTEGRAL CLOSURE A. SEIDENBERG Vol. 16, No. 1 November 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966 DERIVATIONS AND INTEGRAL CLOSURE A. SEIDENBERG Let d? be an integral domain containing the rational num- bers, Σ its quotient field, D a derivation of Σ, and &1 the ring of elements in Σ quasi-integral over &. It is shown that if £? then Dέ?f c &'. According to a lemma of Posner [4], which is also used by him in a subsequent paper [5], if 6? is a finite integral domain over a ground field F of characteristic 0 and D is a derivation over F sending έ? into itself, then D also sends the integral closure of έ? into itself. The proof of this in [4] is wrong, but the statement itself is correct and a proof is here supplied. More generally it is proved that if & is any integral domain containing the rational numbers and D is a derivation such that Zλ^c^, then Dέf'aέ?', where &' is the ring of elements in the quotient field Σ of έ? that are quasi-integral over <^. The theorem is not true for characteristic p Φ 0, but if one uses the Hasse-Schmidt differentiations instead of derivations, one gets the corresponding theorem for a completely arbitrary integral domain £?. Let & be an arbitrary integral domain containing the rational numbers, and let & be the integral closure of d?. The question whether D& c & implies Dέ?c έ? is related to the question whether the ring of formal power series <^[[t]] is integrally closed. -
Jacobson Radical and on a Condition for Commutativity of Rings
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 4 Ver. III (Jul - Aug. 2015), PP 65-69 www.iosrjournals.org Jacobson Radical and On A Condition for Commutativity of Rings 1 2 Dilruba Akter , Sotrajit kumar Saha 1(Mathematics, International University of Business Agriculture and Technology, Bangladesh) 2(Mathematics, Jahangirnagar University, Bangladesh) Abstract: Some rings have properties that differ radically from usual number theoretic problems. This fact forces to define what is called Radical of a ring. In Radical theory ideas of Homomorphism and the concept of Semi-simple ring is required where Zorn’s Lemma and also ideas of axiom of choice is very important. Jacobson radical of a ring R consists of those elements in R which annihilates all simple right R-module. Radical properties based on the notion of nilpotence do not seem to yield fruitful results for rings without chain condition. It was not until Perlis introduced the notion of quasi-regularity and Jacobson used it in 1945, that significant chainless results were obtained. Keywords: Commutativity, Ideal, Jacobson Radical, Simple ring, Quasi- regular. I. Introduction Firstly, we have described some relevant definitions and Jacobson Radical, Left and Right Jacobson Radical, impact of ideas of Right quasi-regularity from Jacobson Radical etc have been explained with careful attention. Again using the definitions of Right primitive or Left primitive ideals one can find the connection of Jacobson Radical with these concepts. One important property of Jacobson Radical is that any ring 푅 can be embedded in a ring 푆 with unity such that Jacobson Radical of both 푅 and 푆 are same. -
Basically Full Ideals in Local Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Algebra 250, 371–396 (2002) provided by Elsevier - Publisher Connector doi:10.1006/jabr.2001.9099, available online at http://www.idealibrary.com on Basically Full Ideals in Local Rings William J. Heinzer Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 and Louis J. Ratliff Jr. and David E. Rush Department of Mathematics, University of California, Riverside, California 92521 Communicated by Craig Huneke Received September 4, 2001 Let A be a finitely generated module over a (Noetherian) local ring R M.We say that a nonzero submodule B of A is basically full in A if no minimal basis for B can be extended to a minimal basis of any submodule of A properly containing B. We prove that a basically full submodule of A is M-primary, and that the fol- lowing properties of a nonzero M-primary submodule B of A are equivalent: (a) B is basically full in A; (b) B =MBA M; (c) MB is the irredundant intersection of µB irreducible ideals; (d) µC≤µB for eachcover C of B. Moreover, if B ∗ is an M-primary submodule of A, then B = MBA M is the smallest basically full submodule of A containing B and B → B∗ is a semiprime operation on the set of nonzero M-primary submodules B of A. We prove that all nonzero M-primary ideals are closed withrespect to thisoperation if and only if M is principal. In rela- tion to the closure operation B → B∗, we define and study the bf-reductions of an M-primary submodule D of A; that is, the M-primary submodules C of D suchthat C ⊆ D ⊆ C∗.IfGM denotes the form ring of R withrespect to M and G+M its maximal homogeneous ideal, we prove that Mn =Mn∗ for all (resp. -
Survey on Locally Factorial Krull Domains Publications Du Département De Mathématiques De Lyon, 1980, Tome 17, Fascicule 1 , P
PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON ALAIN BOUVIER Survey on Locally Factorial Krull Domains Publications du Département de Mathématiques de Lyon, 1980, tome 17, fascicule 1 , p. 1-33 <http://www.numdam.org/item?id=PDML_1980__17_1_1_0> © Université de Lyon, 1980, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Publications du Département de Mathématiques Lyon 1980 t. 17-1 SURVEY ON LOCALLY FACTORIAL KRULL DOMAINS Alain Bouvier In the following diagram, we consider some classes of rings which have been introduced to study integrally closed domains: Discrète valuation ring •^^ Noetherians S (DVR) '•^N. non ' \. χ . _ noeterian^^ -, . / Dedekind valuation ^ . ^/ ι \ Factorial u + ^ (UFD) * Regular ι bezout y ^-^ST ><Γ*" \ ^^^^ / locally ^ \ ηηΤΛ , . \( factorial \ GCD-domain s ,( -,Λ , . \ ^ νKrull domains Prîif er - \ ν ^ Krull \ \ f locally " ^N domains \ | factorial j ι -x \ ν domains \ ' i/^ completely j locally % integrally î GCD-domains / closed Integrally closed 1 Survey on locally factorial Krull domains Assuming furthermore that the domains are noetherian the above diagram collapses and becomes the diagram below· Valuation = DVR l Principal = Bezout UFD = GCD Prufer = Dedekind Regular locally GCD = locally factorial = locally factorial Krull l Krull = integrally closed In this note we shall prove some interesting properties of locally factorial Krull domains; namely: 1) They are the Krull domains such that every divisor- ial idéal Is invertible; in comparison, Dedekind domain have the property that non-zero ideals are invertible and UFD1s that every divisorial is principal. -
Equivariant Class Group. I. Finite Generation of the Picard And
Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring Mitsuyasu Hashimoto Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya 464–8602 JAPAN [email protected] Abstract The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. In particular, we prove the following. Let k be a field, G a smooth k-group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G-scheme. Assume that there is a k-scheme Z of finite type and a dominating k-morphism Z → X. Let ϕ : X → Y be a G-invariant morphism such that arXiv:1309.2367v1 [math.AC] 10 Sep 2013 G OY → (ϕ∗OX ) is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X) is finitely generated, then Cl(G, X) and Cl(Y ) are also finitely generated, where Cl(G, X) is the equivariant class group. In fact, Cl(Y ) is a subquotient of Cl(G, X). For actions of con- nected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G. The proof depends on a similar result on (equivariant) Picard groups. 2010 Mathematics Subject Classification. Primary 13A50; Secondary 13C20. Key Words and Phrases. -
A Prime Ideal Principle in Commutative Algebra
Journal of Algebra 319 (2008) 3006–3027 www.elsevier.com/locate/jalgebra A Prime Ideal Principle in commutative algebra T.Y. Lam ∗, Manuel L. Reyes Department of Mathematics, University of California, Berkeley, CA 94720, USA Received 11 May 2007 Available online 18 September 2007 Communicated by Luchezar L. Avramov Abstract In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. © 2007 Elsevier Inc. All rights reserved. Keywords: Commutative algebra; Commutative rings; Prime ideals; Ideal families; Prime ideal principles; Module categories 1. Introduction One of the most basic results in commutative algebra, given as the first theorem in Kaplansky’s book [Ka2], is (1.1) below, which guarantees that certain kinds of ideals in a commutative ring are prime. (In the following, all rings are assumed to be commutative with unity, unless otherwise specified.) * Corresponding author. E-mail addresses: [email protected] (T.Y. -
Arxiv:1507.04134V1 [Math.RA]
NILPOTENT, ALGEBRAIC AND QUASI-REGULAR ELEMENTS IN RINGS AND ALGEBRAS NIK STOPAR Abstract. We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial px with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the K¨othe conjecture, namely the integral rings. Key Words: π-algebraic element, nil ring, integral ring, quasi-regular element, Jacobson radical, upper nilradical 2010 Mathematics Subject Classification: 16N40, 16N20, 16U99 1. Introduction Let R be an associative ring or algebra. Every nilpotent element of R is quasi-regular and algebraic. In addition the quasi-inverse of a nilpotent element is a polynomial in this element. In the first part of this paper we will be interested in the connections between these three notions; nilpo- tency, algebraicity, and quasi-regularity. In particular we will investigate how close are algebraic elements to being nilpotent and how close are quasi- regular elements to being nilpotent. We are motivated by the following two questions: Q1. Algebraic rings and algebras are usually thought of as nice and well arXiv:1507.04134v1 [math.RA] 15 Jul 2015 behaved. For example an algebraic algebra over a field, which has no zero divisors, is a division algebra. On the other hand nil rings and algebras, which are of course algebraic, are bad and hard to deal with. -
Regular Ideals in Commutative Rings, Sublattices of Regular Ideals, and Prtifer Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 111, 404426 (1987) Regular Ideals in Commutative Rings, Sublattices of Regular Ideals, and Prtifer Rings D. D. ANDERSON* AND J. PASCUAL+ Departmenr of Muthematics, The University qf Iowa, Iowa City, Iowa 52242 Communicated by J. DieudonnC Received April 9, 1986 In this paper we investigate regularly generated. regular, semiregular, and faithful ideals in a commutative ring R and the sublattices they determine. Connections with multiplicative lattice theory are given. Given a Priifer ring R we show that there is a Priifer domain D with the sublattice of regular ideals of R isomorphic to the lattice of ideals of D. Numerous examples of rings with zero divisors having cer- tain properties are given. A Priifer ring with an invertible ideal that is not generated by regular elements is constructed. An example is given to show that the intersec- tion of two regular principal ideals need not be generated by regular elements. (I3 1987 Academic Press. lnc 1. INTRODUCTION Throughout this paper, R will be a commutative ring with identity. We are interested in how parts of the multiplicative theory of ideals for integral domains extend to rings with zero divisors; in particular, to what extent the regular ideals of a ring R behave like the ideals of an integral domain. Section 2 contains the necessary definitions and some preliminary remarks. Regularly generated, regular, semiregular, and faithful ideals are defined along with several “regularity” conditions for rings. A brief outline of the method of idealization is given. -
Idempotent Pairs and PRINC Domains
Idempotent pairs and PRINC domains Giulio Peruginelli, Luigi Salce and Paolo Zanardo∗ August 28, 2018 Dedicated to Franz Halter-Koch on the occasion of his 70th birthday Abstract A pair of elements a,b in an integral domain R is an idempotent pair if either a(1 a) bR, − ∈ or b(1 b) aR. R is said to be a PRINC domain if all the ideals generated by an idempotent − ∈ pair are principal. We show that in an order R of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if R is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders Z[√ d], d > 0 − square-free, that are PRINC and not integrally closed, are for d = 3, 7. Keywords: Orders, Conductor, Primary decomposition, Dedekind domains, Principal ideals, Projective- free. 2010 Mathematics Subject Classification: 13G05, 13F05, 13C10, 11R11. Introduction. Let R be an integral domain, Mn(R) the ring of matrices of order n with entries in R, T any singular matrix in Mn(R) (i.e. det T = 0). A natural question is to find conditions on R to ensure that T is always a product of idempotent matrices. This problem was much investigated in past years, see [17] for comprehensive references. The case when R is a B´ezout domain (i.e. the finitely generated ideals of R are all principal) is crucial. In fact, we recall three fundamental results, valid for matrices with entries in a B´ezout domain. -
Jacobson Radical and Nilpotent Elements
East Asian Math. J. Vol. 34 (2018), No. 1, pp. 039{046 http://dx.doi.org/10.7858/eamj.2018.005 JACOBSON RADICAL AND NILPOTENT ELEMENTS Chan Huh∗ y, Jeoung Soo Cheon, and Sun Hye Nam Abstract. In this article we consider rings whose Jacobson radical con- tains all the nilpotent elements, and call such a ring an NJ-ring. The class of NJ-rings contains NI-rings and one-sided quasi-duo rings. We also prove that the Koethe conjecture holds if and only if the polynomial ring R[x] is NJ for every NI-ring R. 1. Introduction Throughout R denotes an associative ring with identity unless otherwise stated. An element a 2 R is nilpotent if an = 0 for some integer n ≥ 1, and an (one-sided) ideal is nil if all the elements are nilpotent. R is reduced if it has no nonzero nilpotent elements. For a ring R, Nil(R), N(R), and J(R) denote the set of all the nilpotent elements, the nil radical, and the Jacobson radical of R, respectively. Note that N(R) ⊆ Nil(R) and N(R) ⊆ J(R). Due to Marks [14], R is called an NI-ring if Nil(R) ⊆ N(R) (or equilvalently N(R) = Nil(R)). Thus R is NI if and only if Nil(R) forms an ideal if and only if the factor ring R=N(R) is reduced. Hong et al [8, corollary 13] proved that R is NI if and only if every minimal strongly prime ideal of R is completely prime.