Identity-preserving embeddings of countable rings into 2-generator rings

O’Meara, K. C.; Vinsonhaler, C.; Wickless, W.

doi:10.1216/rmj-1989-19-4-1095

Cited by 8 publications

(10 citation statements)

References 4 publications

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“…As an immediate corollary, we will show that every countable ring can be embedded in a ring generated by two elements, reproducing a result of Maltsev (cf. [9] and [10] for different proofs). The group-theoretic analog of this fact has also been known for a long time (cf.…”

Section: Introductionmentioning

confidence: 99%

Endomorphism rings generated using small numbers of elements

Mesyan

2007

Bulletin of the London Mathematical Society

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Let R be a ring, M a nonzero left R-module, Ω an infinite set, and E = End R ( Ω M ). Given two subringsWe show that if M is simple and Ω is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of endomorphisms that take each copy of M to itself) fall into exactly two equivalence classes, with respect to the equivalence relation above. We also show that every countable subset of E is contained in a 2-generator subsemigroup of E.

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Section: Introductionmentioning

confidence: 99%

Endomorphism rings generated using small numbers of elements

Mesyan

2007

Bulletin of the London Mathematical Society

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“…The second result is the fact that any countable-dimensional algebra can be embedded in some finitely generated algebra [3]. We stress that all our algebra embeddings are required to preserve the identity.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

A new measure of growth for countable-dimensional algebras

Hannah¹,

O’Meara²

1993

Bull. Amer. Math. Soc.

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Abstract.A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values for finitely generated algebras exactly fill the unit interval [0,1 ]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite GK-dimension.

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“…Perhaps somewhat surprisingly, finitely generated infinite simple rings seem less ubiquitous. O'Meara et al [20] deduce the existence of such rings with identity as a consequence of their embedding theorem. We construct an example of a finitely generated infinite simple ring without identity in the following section.…”

Section: Remark 52 Ifmentioning

confidence: 96%

Growth of Generating Sets for Direct Powers of Classical Algebraic Structures

Quick

Ruškuc

2010

J. Aust. Math. Soc.

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Communicated by M. G. JacksonDedicated to the memory of Jim Wiegold. AbstractFor an algebraic structure A denote by d( A) the smallest size of a generating set for A, and let, where A n denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d( A) when A is one of the classical structures-a group, ring, module, algebra or Lie algebra. We show that if A is finite then d( A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d( A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.2000 Mathematics subject classification: primary 08A40; secondary 16S15, 17B99, 20F05.

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Identity-preserving embeddings of countable rings into 2-generator rings

Cited by 8 publications

References 4 publications

Endomorphism rings generated using small numbers of elements

Endomorphism rings generated using small numbers of elements

A new measure of growth for countable-dimensional algebras

Growth of Generating Sets for Direct Powers of Classical Algebraic Structures

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