DOCSLIB.ORG

Baer Rings and Baer *-Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Baer Rings and Baer *-Rings BAER RINGS AND BAER *-RINGS S. K. Berberian The University of Texas at Austin ii Registered U.S. Copyright Office March 1988 iii FOREWORD The theory of Baer *-rings was set down in definitive form by Irving Kaplansky (its creator) in 1968, but the subject refuses to stop evolving; these notes are an attempt to record the present state of its evolution, taking into account especially the simplifications due to S. Maeda, S.S. Holland and D. Handelman. Also noted are the connections (first explored by J.-E. Roos and G. Renault) with the theory of regular self-injective rings exposed in K. R. Goodearl's book [Von Neumann regular rings, 1979] and with the theory of continuous geometries, maximal rings of quotients, and von Neumann algebras. Kaplansky's axiomatic approach for studying simultaneously the classical equiv- alence relations on projection lattices is developed in detail, culminating in the construction of a dimension function in that context. The foregoing makes plain that this is less a new venture than it is a con- solidation of old debts. I am especially grateful to Professors Maeda, Holland and Handelman for explaining to me several key points in their work; their generous help made it possible for me to comprehend and incorporate into these notes substantial portions of their work. Each time I survey this theory I learn something new. The most important lessons I learned this time are the following: 1. The incisive results of Maeda and Holland on the interrelations of the various axioms greatly simplify and generalize many earlier results. 2. The connection with regular, right self-injective rings puts the theory in a satisfying general-algebraic framework (the ghost of operator theory surviving only in the involution). 3. There is a wealth of new ideas in Handelman's 1976 Transactions paper; it will be awhile before I can absorb it all. 4. Somehow, the touchstone to the whole theory is Kaplansky's proof of direct finiteness for a complete *-regular ring (Annals of Math., 1955). That it is an exhiliarating technical tour de force does not diminish one's yearning for a simpler proof. The lack of one suggests to me that this profound result needs to be better digested by ring theory. Sterling Berberian Poitiers, Spring 1982 Preface to the English version: The original French version (Anneaux et *- anneaux de Baer) was informally available at the University of Poitiers in the Spring of 1982. Apart from the correction of errors (and, no doubt, the introduction of new iv ones), the present version differs from the French only in the addition of a number of footnotes (some of them clarifications, others citing more recent literature). S.K.B. Austin, Texas, January, 1988 Preface to the second English version: The first English version (with a cir- culation of approximately 40) was produced on a 9-pin dot-matrix printer. The present version was produced from a TEX file created using LEO (ABK Software). I am most grateful to Margaret Combs, the University of Texas Mathematics De- partment's virtuoso TEXperson, for gently guiding me through (and around) the fine points of TEX without insisting that I learn it. S.K.B. Austin, Texas, March, 1991 Preface to the third English version: The diagrams on pages 57, 75 and 102, and the block matrices on pages 119{120, have been re-coded so as to avoid hand-drawn elements. S.K.B. Austin, Texas, April, 2003 TABLE OF CONTENTS FOREWORD . iii x1. Rickart rings, Baer rings . 1 x2. Corners . 12 x3. Center . 14 x4. Commutants . 22 x5. Equivalence of idempotents . 26 x6. *-Equivalence of projections . 31 x7. Directly finite idempotents in a Baer ring . 36 x8. Abelian idempotents in a Baer ring; type theory . 40 x9. Abstract type decomposition of a Baer *-ring . 44 x10. Kaplansky's axioms (A-H, etc.): a survey of results . 48 x11. Equivalence and *-equivalence in Baer *-rings: first properties . 52 x12. The parallelogram law (Axiom H) . 54 x13. Generalized comparability . 58 x14. Polar decomposition . 63 x15. Finite and infinite rings . 74 x16. Rings of type I . 80 x17. Continuous rings . 85 x18. Additivity of equivalence . 87 x19. Dimension functions in finite rings . 95 x20. Continuity of the lattice operations . 98 x21. Extending the involution . 104 REFERENCES . 124 INDEX OF TERMINOLOGY . 126 INDEX OF NOTATIONS . 130 vi 1. RICKART RINGS, BAER RINGS By ring we mean ring with unity; a subring B of a ring A is assumed to contain the unity element of A . 1.1. DEFINITION. [23, p. 510] A Rickart ring is a ring such that the right an- nihilator (resp. left annihilator) of each element is the principal right ideal (resp. left ideal) generated by an idempotent. 1.2. Let A be a Rickart ring, x 2 A . Say fxgl = A(1 − e) ; fxgr = (1 − f)A ; e and f idempotents.1 Then yx = 0 , y(1 − e) = y , ye = 0 ; xz = 0 , (1 − f)z = z , fz = 0 ; whence x2 = 0 , fe = 0 . And (1 − e)x = 0 = x(1 − f) , so ex = x = xf . 1.3. If A is a *-ring (that is, a ring with involution) then \right" is enough in Definition 1.1: one has fxgl = (fx∗gr)∗ , so if fx∗gr = gA with g idempotent, then fxgl = Ag∗ . 1.4. DEFINITION. [23, p. 522] A Rickart *-ring is a *-ring such that the right annihilator of each element is the principal right ideal generated by a projection (a self-adjoint idempotent). 1.5. In a Rickart *-ring, all's well with \left": if fx∗gr = gA , where g∗ = g = g2 , then fxgl = (gA)∗ = Ag . 1.6. The projection in 1.4 is unique. fProof: If eA = fA with e and f projections, then fe = e and ef = f , so e = e∗ = (fe)∗ = e∗f ∗ = ef = f .g 1.7. DEFINITION. Let A be a Rickart *-ring, x 2 A , and write fxgl = A(1 − e) ; fxgr = (1 − f)A 1For any subset S of a ring A , Sl = fx 2 A : xs = 0 (8 s 2 S) g , Sr = fx 2 A : sx = 0 (8 s 2 S) g . 1 2 x1. rickart and baer rings with e and f projections. Then e and f are unique (1.6), and (cf. 1.2) yx = 0 , ye = 0 ; xz = 0 , fz = 0 ; ex = x = xf : One writes e = LP(x) , f = RP(x) , called the left projection and the right projection of x . Thus fxgl = A(1 − LP(x)) ; fxgr = (1 − RP(x))A : By 1.5, one has LP(x∗) = RP(x) , RP(x∗) = LP(x) . 1.8. For idempotents e; f in a ring A , one writes e ≤ f in case e 2 fAf , that is, ef = fe = e . For projections e; f in a *-ring A , the following conditions are equivalent: e ≤ f , e = ef , e = fe , eA ⊂ fA , Ae ⊂ Af . 1.9. In a Rickart *-ring A , for an element x 2 A and a projection g 2 A , one has gx = x , g ≥ LP(x) : fProof: If e = LP(x) , then fxgl = A(1 − e) , so gx = x , (1 − g)x = 0 , 1 − g 2 A(1 − e) , 1 − g ≤ 1 − e , e ≤ g .g 1.10. In a Rickart *-ring, the involution is proper, that is, x∗x = 0 ) x = 0 . fProof: If x∗x = 0 then (cf. 1.7) 0 = RP(x∗)LP(x) = LP(x)LP(x) = LP(x) , hence x = LP(x) · x = 0 .g 1.11. PROPOSITION. [14] Let A be a *-ring. The following conditions are equivalent: (a) A is a Rickart *-ring; (b) A is a Rickart ring and Ae = Ae∗e for every idempotent e . Proof . (a) ) (b): Let e 2 A be idempotent. Then Ae = f1 − egl = Ag with g a projection. One has eg = e and ge = g , whence ege = e and g = g∗ = (ge)∗ = e∗g∗ = e∗g ; therefore e = ege = e(e∗g)e , whence eA ⊂ ee∗A . But ee∗A ⊂ eA trivially, so eA = ee∗A . Taking adjoints, Ae∗ = Aee∗ ; applying this to e∗ in place of e , one obtains Ae = Ae∗e . (b) ) (a): Let e 2 A be idempotent. By (b), e = ae∗e for suitable a . Set f = ae∗ . Then ff ∗ = (ae∗)(ea∗) = (ae∗e)a∗ = ea∗ = f ∗ ; whence f is self-adjoint and f 2 = f . From f = ae∗ one has Af ⊂ Ae∗ ; from e = (ae∗)e = fe one has e∗ = e∗f , so Ae∗ ⊂ Af ; thus Ae∗ = Af . Now let x 2 A be arbitrary. Write fxgl = Ag , g idempotent. Set e = g∗ ; by the preceding, there exists a projection f such that Af = Ae∗ = Ag = fxgl , thus A is a Rickart *-ring. } 1.12. EXAMPLE. Every (von Neumann) regular ring [7, p. 1] is a Rickart ring. fProof: Let A be a regular ring, x 2 A . Choose y 2 A with x = xyx and let e = xy ; then e is idempotent, xA = eA , and fxgl = (xA)l = (eA)l = x1. rickart and baer rings 3 A(1 − e) . Similarly, f = yx is idempotent, Ax = Af , and fxgr = (1 − f)A . Incidentally, fxglr = (A(1 − e))r = f1 − egr = eA = xA ; similarly fxgrl = Ax ; thus (xA)lr = xA and (Ax)rl = Ax .g 1.13. PROPOSITION. [27, p. 114, Th. 4.5] The following conditions on a *-ring A are equivalent: (a) A is a regular Rickart *-ring; (b) A is regular and the involution is proper; (c) for every x 2 A there exists a projection e such that xA = eA . Proof . (a) ) (b): Every Rickart *-ring has proper involution (1.10). (b) ) (a): By 1.12, A is a Rickart ring.
Load more

Recommended publications
  • Arxiv:2005.02059V1 [Math.RA]
    A new approach to Baer and dual Baer modules with some applications N. Ghaedan 1 and M.R. Vedadi 2 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, IRAN. ABSTRACT. Let R be a ring. It is proved that an R-module M is Baer (resp. dual Baer) if and only if every exact sequence 0 → X → M → Y → 0 with Y ∈ Cog(MR) (resp. X ∈ Gen(MR)) splits. This shows that being (dual) Baer is a Morita invariant property. As more + applications, the Baer condition for the R-module M = HomZ(M, Q/Z) is investigated and shown that R is a von Neumann regular ring, if R+ is a Baer R-module. Baer modules with (weak) chain conditions are studied and determined when a Baer (resp. dual baer) module is a direct sum of mutually orthogonal prime (resp. co-prime) modules. While finitely generated dual Baer modules over commutative rings is shown to be semisimple, finitely generated Baer modules over commutative domain are studied. In particular, if R is commutative hereditary Noetherian domain then a finitely generated MR is Baer if and only if it is projective or semisim- ple. Over a right duo perfect ring, it is shown that every (dual) Baer modules is semisimple. Keywords: Baer module, character module, co-prime module, prime module, dual baer, regular ring, retractable module. MSC(2010): Primary: 16D10; 16D40 Secondary: 13C05; 13C10. 1. Introduction Throughout rings will have unit elements and modules will be right unitary. A ring R is said to be Baer if for every non-empty subset X of R, the right annihilator X in R is of the form eR for some e = e2 ∈ R.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • SS-Injective Modules and Rings Arxiv:1607.07924V1 [Math.RA] 27
    SS-Injective Modules and Rings Adel Salim Tayyah Department of Mathematics, College of Computer Science and Information Technology, Al-Qadisiyah University, Al-Qadisiyah, Iraq Email: [email protected] Akeel Ramadan Mehdi Department of Mathematics, College of Education, Al-Qadisiyah University, P. O. Box 88, Al-Qadisiyah, Iraq Email: [email protected] March 19, 2018 Abstract We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R- injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provide some new characterizations of universally mininjective rings, quasi-Frobenius rings, Artinian rings and semisimple rings. Key words and phrases: Small injective rings (modules); soc-injective rings (modules); SS- Injective rings (modules); Perfect rings; quasi-Frobenius rings. 2010 Mathematics Subject Classification: Primary: 16D50, 16D60, 16D80 ; Secondary: 16P20, 16P40, 16L60 . arXiv:1607.07924v1 [math.RA] 27 Jul 2016 ∗ The results of this paper will be part of a MSc thesis of the first author, under the supervision of the second author at the University of Al-Qadisiyah. 1 Introduction Throughout this paper, R is an associative ring with identity, and all modules are unitary R- modules.
    [Show full text]
  • D-Extending Modules
    Hacettepe Journal of Hacet. J. Math. Stat. Volume 49 (3) (2020), 914 – 920 Mathematics & Statistics DOI : 10.15672/hujms.460241 Research Article D-extending modules Yılmaz Durğun Department of Mathematics, Çukurova University, Adana, Turkey Abstract A submodule N of a module M is called d-closed if M/N has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through non- singular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands (respectively, closed, pure). The structure of a ring over which d-closed submodules of every module are direct summand (respectively, closed, pure) is studied. Mathematics Subject Classification (2010). 16D10, 16D40 Keywords. D-extending modules, d-closed submodules, semiartinian modules 1. Introduction Throughout we shall assume that all rings are associative with identity and all modules are unitary right modules. Let R be a ring and let M be an R-module. A (proper) submodule N of M will be denoted by (N M) N ≤ M. By E(M), Rad(M) and Soc(M) we shall denote the injective hull, the Jacobson radical, and the socle of M as usual. For undefined notions used in the text, we refer the reader to [2,11]. A submodule N of a module M is essential (or large) in M, denoted N ¢M, if for every 0 ≠ K ≤ M, we have N ∩ K ≠ 0; and N is said to be closed in M if N has no proper essential extension in M. We also say in this case that N is a closed submodule.
    [Show full text]
  • Virtually Semisimple Modules and a Generalization of the Wedderburn
    Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem ∗†‡ M. Behboodia,b,§ A. Daneshvara and M. R. Vedadia aDepartment of Mathematical Sciences, Isfahan University of Technology P.O.Box : 84156-83111, Isfahan, Iran bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM) P.O.Box : 19395-5746, Tehran, Iran [email protected] [email protected] [email protected] Abstract By any measure, semisimple modules form one of the most important classes of mod- ules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module RM is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) vir- tually semisimple if RR is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple arXiv:1603.05647v2 [math.RA] 31 May 2016 rings. For instance, it is shown that a ring R is left completely virtually semisimple if ∼ k and only if R = Qi=1 Mni (Di) where k,n1, ..., nk ∈ N and each Di is a principal left ideal domain. Moreover, the integers k, n1, ..., nk and the principal left ideal domains D1, ..., Dk are uniquely determined (up to isomorphism) by R.
    [Show full text]
  • Finitely Presented Modules Over Right Non-Singular Rings
    REND.SEM.MAT.UNIV.PADOVA, Vol. 120 2008) Finitely Presented Modules over Right Non-Singular Rings. ULRICH ALBRECHT *) ABSTRACT - Thispaper characterizesthe right non-singularrings R for which M=ZM) isprojective whenever M isa cyclically finitely) presentedmodule. Several related results investigate right semi-hereditary rings. 1. Introduction. The straightforward attempt to extend the notion of torsion-freeness from integral domainsto non-commutative ringsencountersimmediate difficulties. To overcome these, one can concentrate on either the compu- tational or the homological propertiesof torsion-free modules.Goodearl and otherstook the firstapproach when they introduced the notion of a non- singular module [8]. A right R-module M is non-singular if ZM) 0 where ZM) fx 2 M j xI 0 for some essential right ideal I of R} denotesthe singular submodule of M. On the other hand, M is singular if ZM) M. Moreover, a submodule U of an R-module M is S-closed if M=U isnon- singular. Finally, R isa right non-singular ring if RR isnon-singular. The right non-singular rings are precisely the rings which have a regular, right self-injective maximal right ring of quotients, which will be denoted by Qr see [8] and [11] for details). Following [11, Chapter XI], Qr is a perfect left localization of R if Qr isflat asa right R-module and the r r r r multiplication map Q R Q ! Q is an isomorphism. In particular, Q isa perfect left localization of R if and only if every finitely generated non- singular right R-module can be embedded into a projective module [8] and [11]).
    [Show full text]
  • Rings Over Which Certain Modules Are Injective
    Pacific Journal of Mathematics RINGS OVER WHICH CERTAIN MODULES ARE INJECTIVE ANN K. BOYLE AND KENNETH R. GOODEARL Vol. 58, No. 1 March 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 58, No. 1, 1975 RINGS OVER WHICH CERTAIN MODULES ARE INJECTIVE ANN K. BOYLE AND K. R. GOODEARL This paper is concerned with rings for which all modules in one of the following classes are injective: simple modules, quasi-injective modules, or proper cyclic modules. Such rings are known as V-rings, Ql-rings, and PCί-rings, respectively. First, some conditions are developed under which the properties of being a V-ring, QJ-ring, or PCI-ring are left-right symmetric. In the next section, it is shown that a semiprime Goldie ring is a Ql ring if and only if all singular quasi-injective modules are injective. An example is con- structed to show that the class of QJ-rings is properly contained in the class of noetherian V-rings. Also, it is shown that the global homological dimension of a QJ-ring cannot be any larger than its Krull dimension. In the final section, it is shown that a V-ring is noetherian if and only if it has a Krull dimension. Examples are put forward to show that a noether- ian V-ring may have arbitrary finite Krull dimension. 1. Introduction and definitions. A ring R is said to be a right V-ring provided all simple right R -modules are injective. Accord- ing to Villamayor [16, Theorem 2.1], this is equivalent to the condition that every right ideal of R is an intersection of maximal right ideals.
    [Show full text]
  • ON SOME PARTIAL ORDERS on a CERTAIN SUBSET of the POWER SET of RINGS Gregor Dolinar, Bojan Kuzma, Janko Marovt and Burcu Ungor U
    GLASNIK MATEMATICKIˇ Vol. 55(75)(2020), 177 – 190 ON SOME PARTIAL ORDERS ON A CERTAIN SUBSET OF THE POWER SET OF RINGS Gregor Dolinar, Bojan Kuzma, Janko Marovt and Burcu Ungor University of Ljubljana, University of Primorska, University of Maribor, Slovenia and Ankara University, Turkey Abstract. Let R be a ring with identity and let JR be a collection of subsets of R such that their left and right annihilators are generated by the same idempotent. We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to JR, present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to JR, show that they are indeed partial orders when R is a Baer ∗-ring, and connect them with one-sided sharp and star partial orders. 1. Introduction Let S be a semigroup and a ∈ S. We say that a has an inner generalized inverse a− ∈ S if a = aa−a, and a reflexive generalized inverse a′ ∈ S if a = aa′a, a′ = a′aa′. A semigroup in which every element has an inner generalized inverse is called a regular semigroup. A reflexive generalized inverse of a that commutes with a is called the group inverse of a. The group inverse, which is unique if it exists (see [10]), is denoted by a♯. Let R be a ring and let us denote by G(R) the subset of elements in R which have the group inverse. Let Mn(F) be the ring of all n×n matrices over a field F.
    [Show full text]
  • Von-Neumann-Algebra-Like Rings. Presentation at the Joint Mathematics Meeting, Section on Noncommutative Algebra, Washington
    Von-Neumann-algebra-like rings and the answer to a S. K. Berberian’s question A drama in 3 acts Lia Vaˇs University of the Sciences in Philadelphia Main Characters A damsel in distress: A knight in shinning armor: von Neumann Algebra certain Baer ∗-Ring The story of von Neumann Algebra begins... John von Neumann’s dream – to capture abstractly the concept of an algebra of observables in quantum mechanics. He constructed I non-commutative generalization of Hilbert space/ probability theory. Von Neumann Algebra – the powerful Overcomes the limits of classical Suitable for Hilbert space/probability theory. quantum mechanics. Yields all the types of Still capable of describing non-commutative measures large (infinite in size or that occur in classical theory in degrees of freedom) quantum systems. Corresponds to Dimension function normalized measure in classical probability space. Von Neumann Algebra – in distress ”Von Neumann algebras are blessed with an excess of structure – algebraic, geometric, topological – so much, that one can easily obscure, through proof by overkill, what makes a particular theorem work.” ”If all the functional analysis is stripped away ... what remains should (be) completely accessible through algebraic avenues”. Berberian, S. K. Baer ∗-rings; Springer-Verlag, Berlin-Heidelberg-New York, 1972. The overkill The overkill that Berberian is referring to: a mosquito a machine gun VNA Character Expanded H – Hilbert space B(H) – bounded operators. A von Neumann algebra A is a 1) ∗-closed unital subalgebra of B(H), 2a) equal to its double commutant A00 (where A0 = {x ∈ B(H) | ax = xa for all a ∈ A}) equivalently 2b) weakly closed in B(H).
    [Show full text]
  • Semidistributive Modules and Rings Mathematics and Its Applications
    Semidistributive Modules and Rings Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 449 Semidistributive Modules and Rings by Askar A. Tuganbaev Moscow Power Engineering Institute, Technological Univ t!rsiry, Moscow, Russia SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-6136-0 ISBN 978-94-011-5086-6 (eBook) DOI 10.1007/978-94-011-5086-6 Printed on acid-free paper Ali Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Contents Introduction vii Symbols x 1 Radical:-;, local and semisimple modules 1 1.1 Maximal submodules and the Jacobson radical 1 1.2 Local and uniserial modules .... 6 1.3 Semisimple and Artinian modules. 11 1.4 The prime radical. ........ 21 2 Projective and injective modules 25 2.1 Free and projective modules. 25 2.2 Inje,tive modules 31 2.3 Inje,tive hull . 35 3 Bezout nnd regular modules 47 3.1 Regular modules ...... 47 3.2 Unil-regular rings . 52 3.3 Semilocal rings and distributivity 54 3.4 Strongly regular rings 59 3.5 Bezout rings. ........... 68 4 Continuous and finite-dimensional modules 73 4.1 Closed submodules .
    [Show full text]
  • Abstract. a Module M Over a Ring R Satis Es the Restricted Minimum Condi- Tion (RMC) If M=N Is Artinian for Every Essential Submodule N of M
    ON MODULES AND RINGS WITH RESTRICTED MINIMUM CONDITION M. TAMER KOSAN AND JAN ZEMLI CKA Abstract. A module M over a ring R satis es the restricted minimum condi- tion (RMC) if M=N is Artinian for every essential submodule N of M. A ring R satis es right RMC if RR satis es RMC as a right module. It is proved that (1) a right semiartinian ring R satis es right RMC if and only if R=Soc(R) is Artinian, (2) if a semilocal ring R satis es right RMC and Soc(R) = 0, then R is Noetherian if and only if the socle length of E(R=J(R)) is at most !, and (3) a commutative ring R satis es RMC if and only if R=Soc(R) is Noetherian and every singular module is semiartinian. 1. Introduction Given a module over a ring R, we say that M satis es restricted minimum condition (RMC) if for every essential submodule N of M, the factor module M=N is Artinian. It is easy to see that the class of all modules satisfying RMC is closed under submodules, factors and nite direct sums. A ring R satis es right RMC (or we say it is an RM-ring) if RR satis es RMC as a right module. An integral domain R satisfying the restricted minimum condition is called an RM-domain, i.e. R=I is Artinian for all non-zero ideals I of R (see [4]). Recall that a Noetherian domain has Krull dimension 1 if and only if it is an RM-domain [5, Theorem 1].
    [Show full text]