Linear kompakte Moduln und Ringe

Leptin, Horst

doi:10.1007/bf01180634

Cited by 97 publications

(35 citation statements)

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“…Under the last condition, R is a matrix ring over a profinite local ring. We define (see [9]) a left linearly compact ring to be a topological ring having a fundamental system of neighborhoods of zero consisting of left ideals in which the intersection of every filter basis consisting of closed cosets with respect to left ideals is non-empty. For two idempotents of a ring R, we write e e ′ if ee ′ = e ′ e = e. A non-zero idempotent e is said to be minimal in R if there is no non-zero idempotent e 1 = e such that e 1 e. From [10] we recall that a ring R with 1 is said to be clean provided that every element in R is the sum of an idempotent and a unit.…”

Section: Notation and Conventionsmentioning

confidence: 99%

Idempotents and the multiplicative group of some totally bounded rings

Salim

Tripe

2011

Czech Math J

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Abstract. In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power 2 ℵ0 commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

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Section: Notation and Conventionsmentioning

confidence: 99%

Idempotents and the multiplicative group of some totally bounded rings

Salim

Tripe

2011

Czech Math J

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“…Now (e) follows from [23, Proposition 5.2.9] (and its proof), by applying the foregoing definitions (see also [11,Section 10], [20], [21], [29] and [41]). The details are left to the reader.…”

Section: Preliminaries On Quivers and Path Coalgebrasmentioning

confidence: 99%

“…In [11, Section 10], a description of the pseudocompact K-algebra KQ in terms of Cauchy nets is given, where the path algebra KQ is equipped with a K-precompact topology, which is obviously equivalent to our finite subquiver topology (see also [20] and [21]). …”

Section: Preliminaries On Quivers and Path Coalgebrasmentioning

confidence: 99%

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Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Simson

2007

Colloq. Math.

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Abstract. We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q 0 , Q 1 ), we show that any subcoalgebra C of the path K-coalgebra K Q containing K Q 0 ⊕ K Q 1 is the path coalgebra K (Q, B) of a profinite bound quiver (Q, B), and the category C-Comod of left C-comodules is equivalent to the category Rep (M, B), up to isomorphism, is also discussed.

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“…In Halmos' notation alglat M becomes alg (lat λ(R)) and R M K is a reflexive bimodule exactly when λ(R) is a reflexive algebra of operators on R M K . It is of interest to note that Leptin [5] considered the concept of alglat, although not in this notation, for topological modules in a study of completeness and linear compactness. A central lemma in the paper by Hadwin [4] states that if K is an infinite field and R = K[x], the polynomial ring, then the algebra R R K is reflexive as an R, K-bimodule.…”

Section: Introductionmentioning

confidence: 99%

Skew Polynomials and Algebraic Reflexivity

Snashall

Watters

2003

Glasgow Math. J.

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Abstract. For an arbitrary K-algebra R, an R, K-bimodule M is algebraically reflexive if the only K-endomorphisms of M leaving invariant every R-submodule of M are the scalar multiplications by elements of R. Hadwin has shown for an infinite field K and R = K[x] that R is reflexive as an R, K-bimodule. This paper provides a generalisation by giving a skew polynomial version of his result.2000 Mathematics Subject Classification. Primary 16D20, 16S36. Introduction.If V is a vector space over a field K and L is a lattice of subspaces then alg L is defined to be the algebra of all K-endomorphisms of V which leave L point-wise fixed; dually if R is a subalgebra of End K V , then lat R is defined to be the lattice of all subspaces of V which are left invariant by every element of R. Combining these two functors produces alg (lat R), an algebra containing R, and R is called reflexive when these two algebras are equal. Thus, an algebra of operators on a vector space is called reflexive when no larger algebra of operators has the same lattice of invariant subspaces. This terminology originated with Halmos [3] although earlier work also explored some of this area; for example, the lattice of invariant subspaces of a single linear transformation was studied in [1].In [2] a start was made on studying an extension of the concept of alg lat where the focus moved to classes of representations, that is R, K-bimodules, of a K-algebra R. For an arbitrary K-algebra R a bimodule R M K is said to be (algebraically) reflexive if, whenever α ∈ End K M is such that αm ∈ Rm for all m ∈ M, then there is an r ∈ R such that α is simply left multiplication by r.More generally, define for rings R, A and bimodule R M A [2]

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Linear kompakte Moduln und Ringe

Cited by 97 publications

References 0 publications

Idempotents and the multiplicative group of some totally bounded rings

Idempotents and the multiplicative group of some totally bounded rings

Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Skew Polynomials and Algebraic Reflexivity

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