The Radical-Annihilator Monoid of a Ring,” 2016
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Single Elements
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 47 (2006), No. 1, 275-288. Single Elements B. J. Gardner Gordon Mason Department of Mathematics, University of Tasmania Hobart, Tasmania, Australia Department of Mathematics & Statistics, University of New Brunswick Fredericton, N.B., Canada Abstract. In this paper we consider single elements in rings and near- rings. If R is a (near)ring, x ∈ R is called single if axb = 0 ⇒ ax = 0 or xb = 0. In seeking rings in which each element is single, we are led to consider 0-simple rings, a class which lies between division rings and simple rings. MSC 2000: 16U99 (primary), 16Y30 (secondary) Introduction If R is a (not necessarily unital) ring, an element s is called single if whenever asb = 0 then as = 0 or sb = 0. The definition was first given by J. Erdos [2] who used it to obtain results in the representation theory of normed algebras. More recently the idea has been applied by Longstaff and Panaia to certain matrix algebras (see [9] and its bibliography) and they suggest it might be worthy of further study in other contexts. In seeking rings in which every element is single we are led to consider 0-simple rings, a class which lies between division rings and simple rings. In the final section we examine the situation in nearrings and obtain information about minimal N-subgroups of some centralizer nearrings. 1. Single elements in rings We begin with a slightly more general definition. If I is a one-sided ideal in a ring R an element x ∈ R will be called I-single if axb ∈ I ⇒ ax ∈ I or xb ∈ I. -
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. -
Rings Whose Quasi-Injective Modules Are Injecttve K
proceedings of the american mathematical society Volume 33, Number 2, June 1972 RINGS WHOSE QUASI-INJECTIVE MODULES ARE INJECTTVE K. A. BYRD Abstract. A ring R is called a V-ring, respectively SSI-ring, respectively QH-ring if simple, respectively semisimple, respectively quasi-injective, right Ä-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian. The rings R which have the property that quasi-injective right R- modules are injective (herein called QII-rings) have been little mentioned in the literature though related classes of rings have been extensively examined. Consider the classes of R-modules consisting of (1) all R- modules, (2) injective R-modules, (3) projective R-modules, (4) quasi- injective R-modules, and (5) quasi-projective R-modules. Of all the nontrivial inclusions one can assume between two of these classes, all but one has been shown to characterize the ring R as either semisimple artinian, quasi-Frobenius, or uniserial. That one excepted is the class of QII-rings whose internal structure, it seems, has not been completely determined. This paper does not attempt a general structure theorem but investi- gates the connection between QII-rings and rings whose semisimple right modules are injective (SSI-rings). -
ON FULLY IDEMPOTENT RINGS 1. Introduction Throughout This Note
Bull. Korean Math. Soc. 47 (2010), No. 4, pp. 715–726 DOI 10.4134/BKMS.2010.47.4.715 ON FULLY IDEMPOTENT RINGS Young Cheol Jeon, Nam Kyun Kim, and Yang Lee Abstract. We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining hs(Matn(R)) = Matn(hs(R)) for any ring R where hs(−) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunow- icz’s simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process. 1. Introduction Throughout this note each ring is associative with identity unless stated otherwise. Given a ring R, denote the n by n full (resp. upper triangular) matrix ring over R by Matn(R) (resp. Un(R)). Use Eij for the matrix with (i, j)-entry 1 and elsewhere 0. Z denotes the ring of integers. A ring (possibly without identity) is called reduced if it has no nonzero nilpotent elements. A ring (possibly without identity) is called semiprime if the prime radical is zero. Reduced rings are clearly semiprime and note that a commutative ring is semiprime if and only if it is reduced. The study of fully idempotent rings was initiated by Courter [2]. -
Idempotent Lifting and Ring Extensions
IDEMPOTENT LIFTING AND RING EXTENSIONS ALEXANDER J. DIESL, SAMUEL J. DITTMER, AND PACE P. NIELSEN Abstract. We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring R for which idempotents lift modulo the Jacobson radical J(R), but idempotents do not lift modulo J(M2(R)). Thus, the property \idempotents lift modulo the Jacobson radical" is not a Morita invariant. We also prove that if I and J are ideals of R for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over I + J. On the positive side, if I and J are enabling ideals in R, then I + J is also an enabling ideal. We show that if I E R is (weakly) enabling in R, then I[t] is not necessarily (weakly) enabling in R[t] while I t is (weakly) enabling in R t . The latter result is a special case of a more general theorem about completions.J K Finally, we give examplesJ K showing that conjugate idempotents are not necessarily related by a string of perspectivities. 1. Introduction In ring theory it is useful to be able to lift properties of a factor ring of R back to R itself. This is often accomplished by restricting to a nice class of rings. Indeed, certain common classes of rings are defined precisely in terms of such lifting properties. For instance, semiperfect rings are those rings R for which R=J(R) is semisimple and idempotents lift modulo the Jacobson radical. -
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. -
SPECIAL CLEAN ELEMENTS in RINGS 1. Introduction for Any Unital
SPECIAL CLEAN ELEMENTS IN RINGS DINESH KHURANA, T. Y. LAM, PACE P. NIELSEN, AND JANEZ STERˇ Abstract. A clean decomposition a = e + u in a ring R (with idempotent e and unit u) is said to be special if aR \ eR = 0. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott-Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution. 1. Introduction For any unital ring R, the definition of the set of special clean elements in R is motivated by the earlier introduction of the following three well known sets: sreg (R) ⊆ ureg (R) ⊆ reg (R): To recall the definition of these sets, let I(a) = fr 2 R : a = arag denote the set of \inner inverses" for a (as in [28]). Using this notation, the set of regular elements reg (R) consists of a 2 R for which I (a) is nonempty, the set of unit-regular elements ureg (R) consists of a 2 R for which I (a) contains a unit, and the set of strongly regular elements sreg (R) consists of a 2 R for which I (a) contains an element commuting with a. -
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]. -
Quick Course on Commutative Algebra
Part B Commutative Algebra MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA I want to cover Chapters VIII,IX,X,XII. But it is a lot of material. Here is a list of some of the particular topics that I will try to cover. Maybe I won’t get to all of it. (1) integrality (VII.1) (2) transcendental field extensions (VIII.1) (3) Noether normalization (VIII.2) (4) Nullstellensatz (IX.1) (5) ideal-variety correspondence (IX.2) (6) primary decomposition (X.3) [if we have time] (7) completion (XII.2) (8) valuations (VII.3, XII.4) There are some basic facts that I will assume because they are much earlier in the book. You may want to review the definitions and theo- rems: • Localization (II.4): Invert a multiplicative subset, form the quo- tient field of an integral domain (=entire ring), localize at a prime ideal. • PIDs (III.7): k[X] is a PID. All f.g. modules over PID’s are direct sums of cyclic modules. And we proved in class that all submodules of free modules are free over a PID. • Hilbert basis theorem (IV.4): If A is Noetherian then so is A[X]. • Algebraic field extensions (V). Every field has an algebraic clo- sure. If you adjoin all the roots of an equation you get a normal (Galois) extension. An excellent book in this area is Atiyah-Macdonald “Introduction to Commutative Algebra.” 0 MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA 1 Contents 1. Integrality 2 1.1. Integral closure 3 1.2. Integral elements as lattice points 4 1.3. -
A GENERALIZATION of BAER RINGS K. Paykan1 §, A. Moussavi2
International Journal of Pure and Applied Mathematics Volume 99 No. 3 2015, 257-275 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v99i3.3 ijpam.eu A GENERALIZATION OF BAER RINGS K. Paykan1 §, A. Moussavi2 1Department of Mathematics Garmsar Branch Islamic Azad University Garmsar, IRAN 2Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University P.O. Box 14115-134, Tehran, IRAN Abstract: A ring R is called generalized right Baer if for any non-empty subset n S of R, the right annihilator rR(S ) is generated by an idempotent for some positive integer n. Generalized Baer rings are special cases of generalized PP rings and a generalization of Baer rings. In this paper, many properties of these rings are studied and some characterizations of von Neumann regular rings and PP rings are extended. The behavior of the generalized right Baer condition is investigated with respect to various constructions and extensions and it is used to generalize many results on Baer rings and generalized right PP-rings. Some families of generalized right Baer-rings are presented and connections to related classes of rings are investigated. AMS Subject Classification: 16S60, 16D70, 16S50, 16D20, 16L30 Key Words: generalized p.q.-Baer ring, generalized PP-ring, PP-ring, quasi Baer ring, Baer ring, Armendariz ring, annihilator 1. Introduction Throughout this paper all rings are associative with identity and all modules are unital. Recall from [15] that R is a Baer ring if the right annihilator of c 2015 Academic Publications, Ltd. -
Over Noncommutative Noetherian Rings
transactions of the american mathematical society Volume 323, Number 2, February 1991 GENERATING MODULES EFFICIENTLY OVER NONCOMMUTATIVE NOETHERIAN RINGS S. C. COUTINHO Abstract. The Forster-Swan Theorem gives an upper bound on the number of generators of a module over a commutative ring in terms of local data. Stafford showed that this theorem could be generalized to arbitrary right and left noethe- rian rings. In this paper a similar result is proved for right noetherian rings with finite Krull dimension. A new dimension function—the basic dimension—is the main tool used in the proof of this result. Introduction The proof of many interesting results in commutative algebra and algebraic 7C-theory are applications of local-global principles. These are results of the following type: if a property is true for all the localisations of an 7?-module M at each prime ideal of 7?, then it is true for M itself. For example, a principle of this kind lies at the heart of Quillen's solution of Serre's conjecture [8]. A further example of a local-global principle is the Forster-Swan Theorem. Suppose that M is a finitely generated right module over a right noetherian ring R. Let us denote the minimal number of generators of M by gR(M). For a commutative ring R, the Forster-Swan Theorem gives an upper bound for gR(M) in terms of local data; more precisely gR(M) < sup{gR (Mp) + Kdim(7?/P) : P a J -prime ideal of 7? } . For a proof of this theorem and some of its applications to commutative algebra and algebraic geometry, see [8]. -
NOTES in COMMUTATIVE ALGEBRA: PART 1 1. Results/Definitions Of
NOTES IN COMMUTATIVE ALGEBRA: PART 1 KELLER VANDEBOGERT 1. Results/Definitions of Ring Theory It is in this section that a collection of standard results and definitions in commutative ring theory will be presented. For the rest of this paper, any ring R will be assumed commutative with identity. We shall also use "=" and "∼=" (isomorphism) interchangeably, where the context should make the meaning clear. 1.1. The Basics. Definition 1.1. A maximal ideal is any proper ideal that is not con- tained in any strictly larger proper ideal. The set of maximal ideals of a ring R is denoted m-Spec(R). Definition 1.2. A prime ideal p is such that for any a, b 2 R, ab 2 p implies that a or b 2 p. The set of prime ideals of R is denoted Spec(R). p Definition 1.3. The radical of an ideal I, denoted I, is the set of a 2 R such that an 2 I for some positive integer n. Definition 1.4. A primary ideal p is an ideal such that if ab 2 p and a2 = p, then bn 2 p for some positive integer n. In particular, any maximal ideal is prime, and the radical of a pri- mary ideal is prime. Date: September 3, 2017. 1 2 KELLER VANDEBOGERT Definition 1.5. The notation (R; m; k) shall denote the local ring R which has unique maximal ideal m and residue field k := R=m. Example 1.6. Consider the set of smooth functions on a manifold M.