Inverse limits of rings and multiplier rings

Abstract.It is well known that a ring R is an exchange ring iff, for any a ∈ R, a − e ∈ (a 2 − a)R for some e 2 = e ∈ R iff, for any a ∈ R, a − e ∈ R(a 2 − a) for some e 2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a − e ∈ (a 2 − a)R for a unique e 2 = e ∈ R. This condition is not leftright symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227-236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.1991 Mathematics Subject Classification. Primary 16U99. Introduction.Rings R are associative with unity. We write J(R) and U(R) for the Jacobson radical and the group of units of R. In [13], Nicholson observed an interesting relation between elements in a ring R,

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“…This follows from the result of Pedersen and Perera [20]. We refer to Ara [2] for the discussion of exchange rings without unity.…”

Section: Extensions Of Rings An Easy Argument Proves the Following Rmentioning

confidence: 59%

A Class of Exchange Rings

Lee

Zhou

2008

Glasgow Math. J.

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“…The purpose of this example is to show that, in the case of X (H), we have a bounded two-sided approximate identity (P L ) L∈Γ consisting of contractive, self-adjoint projections such that P L T = T = T P L eventually for each T ∈ X (H). We note in passing that algebras which contain a net with this property have been studied in a purely algebraic context by Ara and Perera [4, Definition 1.4] and Pedersen and Perera [25,Section 4]. Let Γ denote the set of all closed, separable subspaces of H, ordered by inclusion.…”

Section: Operator Theory Onmentioning

confidence: 99%

A weak∗‐topological dichotomy with applications in operator theory

Kania

Koszmider

Laustsen

2014

Trans London Maths Soc

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Denote by [0, ω 1 ) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C 0 [0, ω 1 ) be the Banach space of scalar-valued, continuous functions which are defined on [0, ω 1 ) and vanish eventually. We show that a weakly * compact subset of the dual space of C 0 [0, ω 1 ) is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval [0, ω 1 ].Using this result, we deduce that a Banach space which is a quotient of C 0 [0, ω 1 ) can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C 0 [0, ω 1 ) and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy-Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on C 0 [0, ω 1 ). As a consequence, we find that this ideal has a bounded left approximate identity, thus resolving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space C 0 [0, ω 1 ) and the operators acting on it.f ∈ C[0, ω 1 ] : f (ω 1 ) = 0 2010 Mathematics Subject Classification. Primary: 46E15, 47L10; Secondary: 03E05, 46B50, 47L20, 54D30.Key words and phrases. Banach space; continuous functions on the first uncountable ordinal interval; scattered space; uniform Eberlein compactness; weak * topology; club set; stationary set; Pressing Down Lemma; ∆-system Lemma; Banach algebra of bounded operators; maximal ideal; bounded left approximate identity.

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“…We know already that these results are relevant for QB-rings as their stable rank is usually different from one. In fact, from the examples we know of (see [8], [10], [28]) the following question is quite pertinent: Question 3.1. Is the stable rank of a (separative) QB-ring always one, two or infinity?…”

Section: Non-stable K-theorymentioning

confidence: 99%

“…This class was introduced by G. K. Pedersen and the authors in [8], and was subsequently studied in [9], [10], [28]. We say that a unital ring R is a QB-ring if any left unimodular row can be reduced in a special way.…”

Section: Introductionmentioning

confidence: 99%

“…The class of QB-rings is, however, much larger than the class of rings with stable rank one, yet they enjoy specially good structural properties. For example, the property of being a QB-ring is stable under matrix formation, passage to corners, surjective pullbacks, suitable direct and inverse limits (see [8], [10], [28]). The behaviour under extensions is considerably more complicated than that for rings with stable rank one, hence an extension matrix can be reduced to a special standard form.…”

Section: Introductionmentioning

confidence: 99%

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Non-stable $K$-theory for $QB$-rings

Ara¹,

Domènech²

2007

MATH. SCAND.

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We study the class of QB-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a QB-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map GL 1 (R) → K 1 (R) is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of K-theoretical nature. We also show that for a reasonably large class of QB-rings that includes the prime ones, separativity always holds.

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Inverse limits of rings and multiplier rings

Abstract: Abstract. It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that these in many cases can be obtained as inverse limits.

Cited by 7 publications

References 31 publications

A Class of Exchange Rings

A Class of Exchange Rings

A weak∗‐topological dichotomy with applications in operator theory

Non-stable $K$-theory for $QB$-rings

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