Over Noncommutative Noetherian Rings
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On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst
Volume-9 • Number-1 Jan -June 2017 pp. 9-13 available online at www.csjournalss.com A UGC Recommended Journal http://ugc.ac.in/journallist vide letter Dated: 28/03/2017 On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst. Prof. D.A.V. College (Lahore), Ambala City Abstract: A group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and satisfy certain axioms while a ring is an algebraic structure with two binary operations namely addition and multiplication. I have tried to discuss prime ideals, prime radical and m- system in this paper. Keywords: Prime ideals, semi-prime ideals, m-system, n-system. 1. INTRODUCTION In this paper, I have tried to explain the concept of prime ideals in an arbitrary ring, Radical of a ring and few properties of m-system. Besides these some theorems and lemma have been raised such as “If A is an ideal in ring R then B(A) coincide with intersection of all prime ideals in R which contain A”. Also some theorems and lemmas based on m-system and n-system have been established. 1.1 Prime Integer: An integer p is said to be prime integer if it has following property that if a and b are integers such that ab is divisible by p then a is divisible by p or b is divisible by p. 1.2 Prime Ideal [1]: An ideal P in ring R is said to be a prime ideal if and only if it has the following property: If A and B are ideals in R such that AB P, then A P or B P 1.3 m-system: A set M of elements of a ring R is said to be an m-system if and only if it has following property: If a, b M, these exists x R such that axb M 1.4 Theorem: If P is an ideal in ring R, then following conditions are equivalent: (i) P is a prime ideal (ii) If a, b R such that aRb P then a P or b P (iii) If (a) and (b) are principal ideals in R such that (a) (b) P then a P or b P. -
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. -
The Algebraic Method in Representation Theory
THE ALGEBRAIC METHOD IN REPRESENTATION THEORY (ENVELOPING ALGEBRAS) Anthony JOSEPH ~) Institut des Hautes Etudes Scientifiques 91440 - BURES-sur-YVETTE (France) ~) Permanent address : Department of Physics and Astronomy, Tel-Aviv University, Ramit Avic, Israel 98 THE ALGEBRAIC METHOD IN REPRESENTATION THEORY ( ENVELOPING ALGEBRAS) I Introduction In Lie algebra theory a basic open problem is to classify all irreducible representations (up to equivalence). For the category of finite dimensional representations the answer is known and is classical. By contrast a full classification for infinite dimensional representations appears to be impossible. For example, this is evidenced by the work of Arnal and Pinczon [2] on s~(2) and by the work of MeConnell and Robson [34] on A 1 which can be used to show [5] that the Heisenberg Lie algebra [18], 4.6.1, admits infinitely many inequivalent irreducible representations all very different from the standard one. One way out of this difficulty is to consider only representations which integrate to unitary (or just bounded) representations of the corresponding Lie group(s). This has physical justification through Wigner's theorem. We call it the analytic approach. In it the Lie algebra plays only a subservient role. About ten years ago, Di~nier proposed a purely algebraic way out which has since then generated a new area of mathematics called enveloping algebras. The philosophy is to find a less refined classification than equivalence classes rather than to exclude representations. We call it the algebraic approach. Except for finite dimensional representations, or for nilpotent Lie algebras, the exact relationship between these two approaches is not yet known, though one can find many analogies. -
Semi-Prime Rings
SEMI-PRIME RINGS BY R. E. JOHNSON Following Nagata [2], we call an ideal of a ring semi-prime if and only if it is an intersection of prime ideals of the ring. A semi-prime ring is one in which the zero ideal is semi-prime. In view of the definition of the prime radical of a ring given by McCoy [l, p. 829], we have that a ring is semi- prime if and only if it has a zero prime radical. The semi-simple rings of Jacobson [6] are also semi-prime. In the first section of this paper, general properties of a semi-prime ring R are developed. The discussion centers around the concept of the component Ie of a right ideal / of R. The component Ie of / is just the left annihilator of the right annihilator of I. The second section is devoted to a study of the prime right ideals of a semi-prime ring R. Prime right ideals are defined in R much as they are in a prime ring [7]. Associated with each right ideal / of R is a least prime right ideal pil) containing I. Some generalizations of results of [7] are obtained. Thus if I and E are any right ideals of R and if a is any element of R, it is proved that piliM') =pil)r\pil') and that pül:a)) = ipil):a). In the third section, the structure of a semi-prime ring R is introduced along the lines of the structure of a prime ring given in [7] and [8]. -
Lectures on Non-Commutative Rings
Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address. -
Hereditary Rings Integral Over Their Centers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 102, 119-128 (1986) Hereditary Rings Integral over Their Centers ELLEN KIRKMAN AND JAMES KUZMANOVICH Department of Mathematics and Computer Science, Wake Forest University, Box 7311, Reynolda Station, Winston-Salem, North Carolina 27109 Communicated by P. M. Cohn Received October 1, 1984 Bergman [2] has completely characterized the center of a right hereditary ring; the center of a right hereditary ring is a Krull p. p. ring, and any Krull p. p. ring is the center of a right hereditary. While the center of a hereditary ring thus need not be hereditary, there are conditions which imply that the center is hereditary. Robson and Small [13] have shown that the center of a prime PI right hereditary ring is a Dedekind domain, and the center of a PI hereditary Noetherian ring is a finite direct sum of Dedekind domains. The center of a right hereditary PI ring need not be hereditary; Small and Wadsworth [16] have given an example of a PI right hereditary, right Noetherian ring whose center is not Noetherian or semihereditary. Jondrup [ 111 showed that a right hereditary ring which is module-finite over its center has a hereditary center. Chatters and Jondrup [S] showed that a PI right hereditary ring which is ring-finite over its cen- ter has a hereditary center. We shall prove that any right hereditary ring integral over its center has a hereditary center. Chatters and Jondrup [5] ask if a right and left hereditary PI ring has always has a hereditary center; we will produce an example to show it does not. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title One-sided prime ideals in noncommutative algebra Permalink https://escholarship.org/uc/item/7ts6k0b7 Author Reyes, Manuel Lionel Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Koushik Sen Spring 2010 One-sided prime ideals in noncommutative algebra Copyright 2010 by Manuel Lionel Reyes 1 Abstract One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplan- sky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bi- jectively to the classes of cyclic right R-modules that are closed under extensions. -
∗-PRIME GROUP RINGS 1. Introduction Let R Be an Associative
December 15, 2009 14:39 WSPC/171-JAA 00364 Journal of Algebra and Its Applications Vol. 8, No. 6 (2009) 797–803 c World Scientific Publishing Company ∗-PRIME GROUP RINGS KANCHAN JOSHI∗,R.K.SHARMA† and J. B. SRIVASTAVA ‡ Department of Mathematics Indian Institute of Technology Delhi New Delhi, India ∗[email protected] †[email protected] ‡[email protected] Received 20 January 2009 Accepted 17 March 2009 Communicated by M. Ferrero In this paper we characterize ∗-prime group rings. We prove that the group ring RG of the group G over the ring R is ∗-prime if and only if R is ∗-prime and Λ+(G) = (1). In the process we obtain more examples of group rings which are ∗-prime but not strongly prime. Keywords: Strongly prime; ∗-prime; prime; group rings. Mathematics Subject Classification 2000: 16S34, 16N60, 20C07 1. Introduction Let R be an associative ring with identity 1 =0and G be a group. It is well known [1] that the group ring RG of the group G over the ring R is prime if and only if R is prime and the FC subgroup ∆(G) is torsion free abelian or equivalently its torsion subgroup ∆+(G) = (1). AringR is said to be (right) strongly prime if for every 0 = r ∈ R,there exists a finite subset X of R such that rXt =0,t∈ R ⇒ t = 0. Left strongly prime rings are defined analogously. These were first studied by Handelman and Lawrence [2]. Clearly, strongly prime rings are prime. For group rings they proved that if RG is strongly prime then R is strongly prime and G has no non-trivial locally finite normal subgroup, that is, the locally finite radical L(G) = (1). -
Do Simple Rings Have Unity Elements?
JOURNAL OF ALGEBRA 7, 140-143 (1967) Do Simple Rings Have Unity Elements? J. C. ROBSON Department of Mathematics, The University, Leeds, England Communicated by A. W. Goldie Received October 20, 1966 1. It is well known that a simple artinian ring has a unity element, and that a commutative simple ring, being a field, has a unity element. However, it can easily be seenthat not every simple ring hasa unity element, this being shown in the first proposition of this note. Nevertheless, it could be asked whether a simple right noetherian ring necessarilyhas a unity element and the answer (which comprisesthe main result of this note) is a qualified “yes”. 2. There are simple integral domains which are not division rings-see, for example, [2]. Thus the following result demonstrates that there exist simple integral domains without unity elements. PROPOSITION. Let R be a simple integral domain which is not a division ring and let I be a proper nonzero right ideal of R. Then IR is a simpleintegral domain without a unity element. Proof. Let T be a nonzero ideal of IR. Then IR = I(RTIR) = (IR) T(IR) C T, and so T = IR and IR is a simple integral domain. Say IR has a unity element e. Then e is an idempotent. Thus ear = er for all r E R, and hence er = r since R is an integral domain. Therefore R=eRCIR*R=IRCI, which contradicts the assumptionthat I was a proper right ideal of R. So IR does not have a unity element. 3. -
Ideals in Topological Rings
IDEALS IN TOPOLOGICAL RINGS BERTRAM YOOD 1. Introduction. We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously iso morphic to a dense subring of a Q-r'mg (for the notion of a Q-rmg see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semi- simple ring with dense socle. Another such problem is a problem of Kaplansky (6) to determine if R is a Qr-r'mg if and only if R is a Qrrmg. This is true for all quasi-Q-rings. These facts suggest the desirability of a systematic investigation of quasi-Q-rings. These rings have some interesting properties not shared by all topological rings. These involve the notion of a maximal-closed modular right (left) ideal (i.e. maximal in the set of all closed modular right (left) ideals). Examples show that this notion differs from that of a closed modular maximal right (left) ideal. If R is a quasi-Q-ring, then every modular right (left) ideal which is not dense is contained in a maximal-closed modular right (left) ideal (but not necessarily in a closed maximal right (left) ideal. -
Arxiv:Math/0202079V4
ON PRIMITIVE IDEALS VICTOR GINZBURG ABSTRACT. We extendtwo well-known resultson primitive idealsin enveloping algebras of semisim- ple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive ideals, to much wider classes of algebras. Our general version of the Irreducibility Theorem says that if A is a positively filtered associative algebra such that gr A is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of the Duflo theorem says that if A is an algebra with a ”triangular structure”, then any primitive ideal in A is the annihilator of a simple highest weight module. Applications to symplectic reflection algebras and Cherednik algebras are discussed. CONTENTS 1. Introduction 1 2. Main results 3 3. Proof of Theorem 2.1.1 5 4. Category ↑O 8 5. Jacquet functor and Proof of Theorem 2.3.1 13 6. Applications to symplectic reflection algebras 18 References 21 1. INTRODUCTION 1.1. Let A be an associative, not necessarily commutative, algebra. Recall that a two-sided ideal I ( A is called primitive if it is the annihilator of a simple left A-module. Primitive ideals often play a role similar to the one played by maximal ideals in the case of commutative algebras. The set of all primitive ideals in A, equipped with Jacobson topology, is called the primitive spectrum of A. It is a fundamental invariant of A analogous to the affine scheme Spec A in the commutative case. -
AJMS-J.Ajmsc.2019.12.004 Proof 233..243
The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/1319-5166.htm Primeness of On the primeness of near-rings near-rings Khalid H. Al-Shaalan Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia 233 Abstract In this paper, we study the different kinds of the primeness on the class of near-rings and we give new characterizations for them. For that purpose, we introduce new concepts called set-divisors, ideal-divisors, etc. Received 11 October 2019 Revised 16 December 2019 and we give equivalent statements for 3-primeness which make 3-primeness looks like the forms of the other Accepted 16 December 2019 kinds of primeness. Also, we introduce a new different kind of primeness in near-rings called K-primeness which lies between 3-primeness and e-primeness. After that, we study different kinds of prime ideals in near- rings and find a connection between them and new concepts called set-attractors, ideal-attractors, etc. to make new characterizations for them. Also, we introduce a new different kind of prime ideals in near-rings called K-prime ideals. Keywords Near-rings, Rings, Primeness, Prime ideals Paper type Original Article 1. Introduction We say that R is a right (left) near-ring if ðR; þÞ is a group, ðR; $Þ is a semigroup and R satisfies the right (left) distributive law. Throughout this paper, R will be a left near-ring. We say that R is an abelian near-ring if x þ y ¼ y þ x for all x; y ∈ R and we say that R is a commutative near-ring if xy ¼ yx for all x; y ∈ R.