Radicals coinciding with the von Neumann regular radical on artinian rings

We study radicals which coincide on artinian rings with Jacobson semisimple rings or equivalently with von Neumann regular rings. Exact lower and upper bounds for strong coincidence are given. For weak coincidence the exact lower bound is that for strong coincidence. We determine the smallest homomorphically closed class which contains all radicals coinciding in the weak sense with the von Neumann regular radical on artinian rings, but we do not know even the existence of the upper bound for weak coincidence. If a radical coincides with the von Neumann regular radical on artinian rings in the strong sense, then (A) is a direct summand inA for every aritian ringA.Research carried out within the Austro-Hungarian Bilateral Intergovernmental Cooperation Program A-31. Research partially supported by Hungarian National Foundations for Scientific Research Grant No. T4265The second author gratefully acknowledges the support of the Carnegie Trust for Universities of Scotland     

Cartan-Eilenberg projective,injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.     

Generalizations of von Neumann regular rings, <Emphasis Type="Italic">PP</Emphasis> rings,and Baer rings

We introduce the notions of IDS modules, IP modules, and Baer* modules, which are new generalizations of von Neumann regular rings, PP rings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baer* modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer* modules.     

On a generalisation of self-injective von Neumann regular rings

Apart from von Neumann regular rings, rings with infinite identities have not been studied in any detail. We take a first step in that direction by obtaining structure theorems for a class of self-injective rings with infinite identities. These extend the main structure theorems for self-injective von Neumann regular rings.

     

Morphic rings and unit regular rings

A ring R is called left morphic if for every a∈R. A left and right morphic ring is called a morphic ring. If M_n(R) is morphic for all n≥1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(xⁿ) is strongly morphic for all n≥1 iff R[x]/(x²) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.     

Von Neumann Betti numbers and Novikov type inequalities

In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

     

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.     

The generating hypothesis in the derived category of a ring

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author (J. Pure Appl. Algebra 208(2), 2007). We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular and therefore does not satisfy the strong form of the generating hypothesis.     

A note on cocycle-conjugate endomorphisms of von Neumann algebras

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type

     

On PP-rings

A ringR is called a leftPP-ring if every principal left ideal is projective, equivalently if the left annihilatorl(a) is generated by an idempotent for allaR. These rings seem first to have been discussed by Hattori [2] and examples include (von Neumann) regular rings and domains (possibly noncommutative). In this note we give a new characterization of leftPP-rings, use that to give an elementary proof of a result of Xue [4] characterizing triangularPP-rings, and then determine when the ringT _n (R) of upper triangular matrices overR is a leftPP-ring. Throughout the paper all rings have a unity and all modules are unitary.This research was supported by NSERC Grant A8075     

Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple Rings and Domains

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L _K (n, n + k) constructed by Leavitt. Using Bergman’s diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.     

A uniform refinement property for congruence lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice with compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice cannot be obtained using the Pudlák, Tischendorf, Tuma approach.

The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

     

On von Neumann regular rings,XIII

Summary Generalizations of projectivity and quasi-injectivity, calledC-projectivity andIC-injectivity, are introduced to study von Neumann regular rings, continuous and self-injetive regular rings. Conditions for non-reduced ideals to contain non-trivial central idempotents are considered.

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Riassunto Vengono introdotte delle generalizzazioni delle proiettività e delle quasi-iniettività detteC-proiettività eIC-iniettività per studiare gli anelli regolari di von Neumann, anelli continui e regolari auto-iniettivi. Sono inoltre considerate condizioni a<nchè ideali non ridotti contengano idempotenti centrali non banali.

     

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K₀-theory), that to a ring R associates the monoid V(?R?) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor:

1. There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V °?Γ?? id.
2. There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V °?Γ?? id.
3. There is a {0,1}³-indexed commutative diagram  ${\vec{D}}$ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0.

By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality  $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(?R?) is the positive cone of a dimension group but it is not isomorphic to V(?B?) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.     

On regular rings and continuous rings,III

Summary In this sequel to [14]and [15],a generalization of quasi-injective modules, noted FK-injective, is introduced to study von Neumann regular and continuous rings. This will lead to new characteristic properties of continuous regular rings. Conditions for certain non-singular modules to be completely reducible and injective are given. A few decompositions of FK-injective modules are considered.Dedicated to Professor Hisao Tominaga     

On transitive algebras containing a standard finite von Neumann subalgebra

Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M^′. In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B(H). This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271-283]) is 2-fold transitive, then A is strongly dense in B(H). Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., LFn and , are studied. Brown measures of certain operators in are explicitly computed.     

On von Neumann regular rings,III

This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.     

A Comparison Between James and von Neumann–Jordan Constants

In this paper, we compare James and von Neumann–Jordan constants of normed spaces under certain conditions. It is shown that if a normed space with James constant \(\sqrt{2}\) is three- or more dimensional, or is a \(\pi /2\)-rotation-invariant two-dimensional space, then its von Neumann–Jordan constant is less than or equal to \(4-2\sqrt{2}\).