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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Gromov translation algebras over discrete trees are exchange rings
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by P. Ara, K. C. O’Meara and F. Perera PDF
Trans. Amer. Math. Soc. 356 (2004), 2067-2079 Request permission

Abstract:

It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras $G(0)$ of $\omega \times \omega$ matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers $r$ in the unit interval $[0,1]$, the growth algebras $G(r)$ (introduced by Hannah and O’Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension $r$ in $[0,1]$.
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Additional Information
  • P. Ara
  • Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain
  • MR Author ID: 206418
  • Email: para@mat.uab.es
  • K. C. O’Meara
  • Affiliation: Department of Mathematics, University of Canterbury, Christchurch, New Zealand
  • Address at time of publication: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
  • Email: K.OMeara@math.canterbury.ac.nz, staf198@ext.canterbury.ac.nz
  • F. Perera
  • Affiliation: Department of Pure Mathematics, Queen’s University Belfast, Belfast, BT7 1NN, Northern Ireland
  • Address at time of publication: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
  • MR Author ID: 620835
  • Email: perera@qub.ac.uk, perera@mat.uab.es
  • Received by editor(s): September 27, 2002
  • Received by editor(s) in revised form: April 15, 2003
  • Published electronically: November 12, 2003
  • Additional Notes: The first and third authors were partially supported by DGESIC, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The initial ideas for this paper were discussed while the second author was visiting the Centre de Recerca Matemàtica, Institut d’Estudis Catalans in Barcelona, and he thanks this institution for its support and hospitality.
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2067-2079
  • MSC (2000): Primary 16E50, 16D70, 16S50
  • DOI: https://doi.org/10.1090/S0002-9947-03-03372-5
  • MathSciNet review: 2031053