Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T02:47:59.008Z Has data issue: false hasContentIssue false
  • English
  • Français

On boolean near-rings

Published online by Cambridge University Press:  17 April 2009

Steve Ligh
Steve Ligh
Affiliation:
University of Florida, Gainesville, Florida.
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.

It is well-known that a boolean ring is commutative. In this note we show that a distributively generated boolean near-ring is multiplicatively commutative, and therefore a ring. This is accomplished by using subdirect sum representations of near-rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 375 - 379
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Beidleman, James C., “Distributively generated near-rings with descending chain condition”, Math. Z. 91 (1966), 65–69.CrossRefGoogle Scholar
[2]Birkhoff, Garrett, “Subdirect unions in universal algebra”, Bull. Amer. Math. Soc. 50 (1944), 764–768.CrossRefGoogle Scholar
[3]Blackett, D.W., “Simple and semisimple near-rings”, Proc. Amer. Math. Soc. 4 (1953), 772–785.CrossRefGoogle Scholar
[4]Clay, James R. and Lawver, Donald A., “Boolean near-rings”, Canad. Math. Bull, (to appear).Google Scholar
[5]Fain, Charles Gilbert, “Some structure theorems for near-rings”, Doctoral thesis, University of Oklahoma, 1968.Google Scholar
[6]Fröhlich, A., “Distributively generated near-rings, (I. Ideal Theory)”, Proc. London Math. Soc. (3) 8 (1958), 76–94.CrossRefGoogle Scholar
[7]Ligh, Steve, “On distributively generated near-rings”, Proc. Edinburgh Math. Soc. (to appear).Google Scholar
[8]Ligh, Steve, “On division near-rings”, Canad. J. Math, (to appear).Google Scholar
[9]Ligh, Steve, “Near-rings with descending chain condition”, Compositio Math, (to appear).Google Scholar
[10]McCoy, N.H. and Montgomery, Deane, “A representation of generalized boolean rings”, Duke Math. J. 3 (1937), 455–459.CrossRefGoogle Scholar
[11]Subrahmanyam, N.V., “Boolean semirings”, Math. Ann. 148 (1962), 395–401.CrossRefGoogle Scholar