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Primitive Near-Rings by William
PRIMITIVE NEAR-RINGS BY WILLIAM MICHAEL LLOYD HOLCOMBE -z/ Thesis presented to the University of Leeds _A tor the degree of Doctor of Philosophy. April 1970 ACKNOWLEDGEMENT I should like to thank Dr. E. W. Wallace (Leeds) for all his help and encouragement during the preparation of this work. CONTENTS Page INTRODUCTION 1 CHAPTER.1 Basic Concepts of Near-rings 51. Definitions of a Near-ring. Examples k §2. The right modules with respect to a near-ring, homomorphisms and ideals. 5 §3. Special types of near-rings and modules 9 CHAPTER 2 Radicals and Semi-simplicity 12 §1. The Jacobson Radicals of a near-ring; 12 §2. Basic properties of the radicals. Another radical object. 13 §3. Near-rings with descending chain conditions. 16 §4. Identity elements in near-rings with zero radicals. 20 §5. The radicals of related near-rings. 25. CHAPTER 3 2-primitive near-rings with identity and descending chain condition on right ideals 29 §1. A Density Theorem for 2-primitive near-rings with identity and d. c. c. on right ideals 29 §2. The consequences of the Density Theorem 40 §3. The connection with simple near-rings 4+6 §4. The decomposition of a near-ring N1 with J2(N) _ (0), and d. c. c. on right ideals. 49 §5. The centre of a near-ring with d. c. c. on right ideals 52 §6. When there are two N-modules of type 2, isomorphic in a 2-primitive near-ring? 55 CHAPTER 4 0-primitive near-rings with identity and d. c. -
Topics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. -
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. -
0.2 Structure Theory
18 d < N, then by Theorem 0.1.27 any element of V N can be expressed as a linear m m combination of elements of the form w1w2 w3 with length(w1w2 w3) ≤ N. By the m Cayley-Hamilton theorem w2 can be expressed as an F -linear combination of smaller n N−1 N N−1 powers of w2 and hence w1w2 w3 is in fact in V . It follows that V = V , and so V n+1 = V n for all n ≥ N. 0.2 Structure theory 0.2.1 Structure theory for Artinian rings To understand affine algebras of GK dimension zero, it is necessary to introduce the concept of an Artinian ring. Definition 0.2.1 A ring R is said to be left Artinian (respectively right Artinian), if R satisfies the descending chain condition on left (resp. right) ideals; that is, for any chain of left (resp. right) ideals I1 ⊇ I2 ⊇ I3 ⊇ · · · there is some n such that In = In+1 = In+2 = ··· : A ring that is both left and right Artinian is said to be Artinian. A related concept is that of a Noetherian ring. Definition 0.2.2 A ring R is said to be left Noetherian (respectively right Noethe- rian) if R satisfies the ascending chain condition on left (resp. right) ideals. Just as in the Artinian case, we declare a ring to be Noetherian if it is both left and right Noetherian. An equivalent definition for a left Artinian ring is that every non-empty collection of left ideals has a minimal element (when ordered under inclusion). -
On the Injective Hulls of Semisimple Modules
transactions of the american mathematical society Volume 155, Number 1, March 1971 ON THE INJECTIVE HULLS OF SEMISIMPLE MODULES BY JEFFREY LEVINEC) Abstract. Let R be a ring. Let T=@ieI E(R¡Mt) and rV=V\isI E(R/Mt), where each M¡ is a maximal right ideal and E(A) is the injective hull of A for any A-module A. We show the following: If R is (von Neumann) regular, E(T) = T iff {R/Mt}le, contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring. Let R be a ring such that every cyclic Ä-module contains a simple. Let {R/Mi]ie¡ be a family of pairwise nonisomorphic simples. Then E(@ts, E(RIMi)) = T~[¡eIE(R/M/). In the commutative regular case these conditions are equivalent. Let R be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form Rlf^\te, Mi, where {Mt}te! is any collection of maximal ideals, contains a simple. We finally look at the relationship between a regular ring R with central idempotents and the Zariski topology on spec R. Introduction. Eckmann and Schopf [2] have shown the existence and unique- ness up to isomorphism of the injective hull of any module. The problem has been to take a specific module and find explicitly its injective hull. -
ON N-FLAT MODULES and N-VON NEUMANN REGULAR RINGS
ON n-FLAT MODULES AND n-VON NEUMANN REGULAR RINGS NAJIB MAHDOU Received 4 May 2006; Revised 20 June 2006; Accepted 21 August 2006 We show that each R-module is n-flat (resp., weakly n-flat) if and only if R is an (n,n − 1)-ring (resp., a weakly (n,n − 1)-ring). We also give a new characterization of n-von Neumann regular rings and a characterization of weak n-von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weak n-von Neumann regular ring is a (CH)-ring. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction All rings considered in this paper are commutative with identity elements and all modules are unital. For a nonnegative integer n,anR-module E is n-presented if there is an exact sequence Fn → Fn−1 →···→F0 → E → 0, in which each Fi is a finitely generated free R- module. In particular, “0-presented” means finitely generated and “1-presented” means finitely presented. Also, pdR E will denote the projective dimension of E as an R-module. Costa [2] introduced a doubly filtered set of classes of rings throwing a brighter light on the structures of non-Noetherian rings. Namely, for nonnegative integers n and d, ≤ we say that a ring R is an (n,d)-ring if pdR(E) d for each n-presented R-module E (as ≤ usual, pd denotes projective dimension); and that R is a weak (n,d)-ring if pdR(E) d for each n-presented cyclic R-module E. -
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. -
Prime Von Neumann Regular Rings and Primitive Group Algebras Joe W
proceedings of the american mathematical society Volume 44, Number 2, June 1974 PRIME VON NEUMANN REGULAR RINGS AND PRIMITIVE GROUP ALGEBRAS JOE W. FISHER AND ROBERT L. SNIDER Abstract. Kaplansky posed the following question: Are prime von Neumann regular rings primitive? We show that with a certain countability condition the answer is affirmative. This is used to sim- plify and clarify earlier work on the existence of primitive group algebras. In [7] Kaplansky posed the following question : Are prime von Neumann regular rings primitive? Our main theorem in §1 asserts that a prime von Neumann regular ring with a countable cofinal subset of ideals is indeed primitive. From this it follows that Kaplansky's question has an affir- mative answer for each of the following classes of rings: (1) countable rings, and (2) rings with the descending chain condition on two-sided ideals. We call locally Artinian a ring which is the union of an ascending sequence of left Artinian subrings. In §2 we establish that a semisimple locally Artinian ring is primitive if and only if it is prime. This elucidates and extends earlier work of Formanek and Snider [5] and Passman [11] on the existence of primitive group algebras. Moreover, we apply it to produce new primitive group algebras by showing that the group algebra over any field of the group of permutations on an infinite set which move only finitely many elements is primitive. 1. Prime von Neumann regular rings. Let R be a prime ring. The unadorned word "ideal" will always mean "two-sided ideal". Under the partial order of inclusion the set S of nonzero ideals of R is a directed set since I,J in S implies /n/in S, InJçI, and inJ^J. -
On Commutatativity of Primitive Rings with Some Identities B
International Journal of Research and Scientific Innovation (IJRSI) | Volume V, Issue IV, April 2018 | ISSN 2321–2705 On Commutatativity of Primitive Rings with Some Identities B. Sridevi*1 and Dr. D.V.Ramy Reddy2 1Assistant Professor of Mathematics, Ravindra College of Engineering for Women, Kurnool- 518002 A.P., India. 2Professor of Mathematics, AVR & SVR College of Engineering And Technology, Ayyaluru, Nandyal-518502, A.P., India Abstract: - In this paper, we prove that some results on (1 + a) (b + ab) + (b + ab) (1 + a) = (1 + a) (b + ba) + ( b + ba) commutativity of primitive rings with some identities + (b + ba ) ( 1 + a) Key Words: Commutative ring, Non associative primitive ring, By simplifying, , Central b + ab + ab + a (ab) + b + ba + ab + (ab) a = b + ba + ab + I. INTRODUCTION a(ba) + b +ba +ba + ba +(ba) a. n this paper, we first study some commutativity theorems of Using 2.1, we get I non-associative primitive rings with some identities in the center. We show that some preliminary results that we need in 2(ab – ba) = 0, i.e., ab = ba. the subsequent discussion and prove some commutativity Hence R is commutative. theorems of non-associative rings and also non-associative primitive ring with (ab)2 – ab 휖 Z(R) or (ab)2 –ba 휖 Z(R) ∀ Theorem 2.2 : If R is a 2 – torsion free non-associative a , b in R is commutative. We also prove that if R is a non- primitive ring with unity 2 2 associative primitive ring with identity (ab) – b(a b) 휖 Z(R) such that (ab)2 – (ba)2 휖 Z(R), for all a, b in R , then R is for all a, b in R is commutative. -
FUSIBLE RINGS 1. Introduction It Is Well-Known That the Sum of Two Zero
FUSIBLE RINGS EBRAHIM GHASHGHAEI AND WARREN WM. MCGOVERN∗ Abstract. A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x; σ] and R[[x; σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semi-simple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full Rings of continuous functions are fusible. It is also shown that if 1 = e1 + e2 + ::: + en in a ring R where the ei are orthogonal idempotents and each eiRei is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings. Keywords: Fusible rings, unit fusible rings, zero-divisors, clean rings, C(X). MSC(2010): Primary:16U99; Secondary:16W99, 13A99. 1. Introduction It is well-known that the sum of two zero-divisors need not be a zero-divisor. In [18, Theorem 1.12] the authors characterized commutative rings for which the set of zero-divisors is an ideal. The set of zero-divisors (i.e., Z(R)) is an ideal in a commutative ring R if and only if the classical ring of quotients of R (i.e., qcl(R)) is a local ring if and only if Z(R) is a prime ideal of R and qcl(R) is the localization at Z(R). -
Generalized V-Rings and Von Neumann Regular Rings
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA GIUSEPPE BACCELLA GeneralizedV-rings and von Neumann regular rings Rendiconti del Seminario Matematico della Università di Padova, tome 72 (1984), p. 117-133 <http://www.numdam.org/item?id=RSMUP_1984__72__117_0> © Rendiconti del Seminario Matematico della Università di Padova, 1984, 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/ Generalized V-Rings and Von Neumann Regular Rings. GIUSEPPE BACCELLA (*) . Introduction and notations. Throughout the present paper all rings are associative with 0 ~ 1 and all modules are unitary. Given a ring .1~, we shall denote with the Jacobson radical of l~. S will be a choosen set of representatives of all simple right R-modules, while P is the subset of S of the representa- tives of all simple projective right R-modules. Given a right R-module M, we shall write for the injective envelope of .~; the notation (resp. will mean that N is an B-submodule (resp. an essential .R-submodule) of For a given subset A c S, we shall denote by SocA the A-homogeneous conponent of the socle Soc ~11 of if; we shall write instead of (M), for a given U E S. -
An Extension of the Jacobson Density Theorem
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 82, Number 4, July 1976 AN EXTENSION OF THE JACOBSON DENSITY THEOREM BY JULIUS ZELMANOWITZ Communicated by Robert M. Fossum, January 26, 1976 The purpose of this note is to outline a generalization of the Jacobson density theorem and to introduce the associated class of rings. Throughout R will be an associative ring, not necessarily possessing an identity element. MR will always denote a right R-module, and homomorphisms will be written on the side opposite to the scalars. For an element m G M, set (0:m) = {rGR\mr = 0}. A nontrivial module MR is called compressible if it can be embedded in any of its nonzero submodules. A compressible module MR is critically com pressible if it cannot be embedded in any proper factor module. For a com pressible module MR, each of the following conditions is equivalent to MR being critically compressible: (1) MR is monoform (i.e. nonzero partial endomorphisms of M are monomorphisms); (2) MR is uniform and nonzero endomorphisms of M are monomorphisms. The proof of these characterizations is elementary. In the absence of more suitable terminology let us define a ring to be weakly primitive if it possesses a faithful critically compressible module. A weakly primitive ring is prime; for it is easy to see that the annihilator of a com pressible module is a prime ideal. In order to simplify this presentation let us call (A, ± VR, MR) an R- lattice if F is a A-R bimodule where A is a division ring, AM = V, and R acts faithfully on M (so that R can be regarded as a subring of End A V).