Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-16T02:55:33.396Z Has data issue: false hasContentIssue false
  • English
  • Français

On zero-dimensionality and fragmented rings

Published online by Cambridge University Press:  17 April 2009

Jim Coykendall ,
David E. Dobbs  and
Bernadette Mullins
Jim Coykendall
Affiliation:
Department of MathematicsNorth Dakota State UniversityFargo ND 58105–5075United States of America
David E. Dobbs
Affiliation:
Department of MathematicsUniversity of TennesseeKnoxville TN 37996–1300United States of America
Bernadette Mullins
Affiliation:
Department of Mathematics and StatisticsYoungstown State UniversityYoungstown OH 44555–3302United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A commutative ring R is said to be fragmented if each nonunit of R is divisible by all positive integral powers of some corresponding nonunit of R. It is shown that each fragmented ring which contains a nonunit non-zero-divisor has (Krull) dimension ∞. We consider the interplay between fragmented rings and both the atomic and the antimatter rings. After developing some results concerning idempotents and nilpotents in fragmented rings, along with some relevant examples, we use the “fragmented” and “locally fragmented” concepts to obtain new characterisations of zero-dimensional rings, von Neumann regular rings, finite products of fields, and fields.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 137 - 151
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Anderson, D.D. and Valdes-Leon, S., ‘Factorization in commutative rings with zero divisors’, Rocky Mountain J. Math. 26 (1996), 439–480.CrossRefGoogle Scholar
[2]Bourbaki, N., Commutative Algebra (Addison-Wesley, Reading, Mass., 1972).Google Scholar
[3]Bourbaki, N., E´léments de mathématique (chapitres 8–9), Algèbre Commutative (Masson, Paris, 1985).Google Scholar
[4]Coykendall, J. and Dobbs, D.E., ‘Fragmented domains have infinite Krull dimension’, (submitted for publication).Google Scholar
[5]Coykendall, J., Dobbs, D.E. and Mullins, B., ‘On integral domains with no atoms’, Comm. Algebra (to appear).Google Scholar
[6]Coykendall, J., Dobbs, D.E. and Mullins, B., ‘Factorization in antimatter rings’, in Advances in Commutative Ring Theory, Lecture Notes Pure Appl. Math. 205 (Dekker, New York, 1999), pp. 217–226.Google Scholar
[7]Dobbs, D.E., ‘Fragmented integral domains’, Portugal. Math. 43 (1985-1986), 463–473.Google Scholar
[8]Huckaba, J.A., Commutative rings with zero divisors (Dekker, New York, 1988).Google Scholar
[9]Kaplansky, I., Commutative Rings, (revised edition) (Polygonal Publishing House, Washington, N.J., 1994).Google Scholar
[10]Lambek, J., Lessons on rings and modules (Blaisdell, Waltham, MA, 1966).Google Scholar
[11]Lewis, W.J., ‘The spectrum of a ring as a partially ordered set’, J. Algebra 25 (1973), 419–434.CrossRefGoogle Scholar
[12]Pierce, R.S., Modules over commutative regular rings, Memoirs Amer. Math. Soc. 70 (American Mathematical Society, Providence, R.I., 1967).Google Scholar
[13]Zariski, O. and Samuel, P., Commutative algebra 1 (Van Nostrand, Princeton, 1958).Google Scholar