Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T01:30:50.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings whose additive endomorphisms are ring endomorphisms

Published online by Cambridge University Press:  17 April 2009

Gary Birkenmeier  and
Henry Heatherly
Gary Birkenmeier
Affiliation:
Department of Mathematics, University of Southwestern, Louisiana Lafayette, Louisiana 70504, United States of America
Henry Heatherly
Affiliation:
Department of Mathematics, University of Southwestern, Louisiana Lafayette, Louisiana 70504, United 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 ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 1 , August 1990 , pp. 145 - 152
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Birkenmeier, G. and Heatherly, H., ‘Left self distributive near-rings’, J. Austral. Math. Soc. (to appear).Google Scholar
[2]Birkenmeier, G., Heatherly, H. and Kepka, T., ‘Rings with left self distributive multiplication’, (submitted).Google Scholar
[3]Dhompongsa, S. and Sanwong, J., ‘Rings in which additive mappings are multiplicative’, Studia Sci. Math. Hungar. 22 (1987), 357–359.Google Scholar
[4]Feigelstock, S., ‘Rings whose additive endomorphisms are multiplicative’, Period. Math. Hungar. 19 (1988), 257–260.CrossRefGoogle Scholar
[5]Feigelstock, S., ‘Rings whose additive endomorphisms are n-multiplicative’, Bull. Austral. Math. Soc. 39 (1989), 11–14.CrossRefGoogle Scholar
[6]Fuchs, L., Infinite Abelian Groups I(Academic Press, New York - London, 1970).Google Scholar
[7]Kaplansky, I., Infinite Abelian Groups (Univ. of Michigan Press, Ann Arbor, Michigan, 1954).Google Scholar
[8]Kim, K.H. and Roush, F.W., ‘Additive endomorphisms of rings’, Period. Math. Hungar. 12 (1981).CrossRefGoogle Scholar
[9]Petrich, M., ‘Structure des demi-groupes et anneaux distributifs’, C.R. Acad. Sci. Paris Sir. A - B (1969), A849–A852.Google Scholar
[10]Sullivan, R.P., ‘Research problems’, Period. Math. Hungar. 8 (1977), 313–314.Google Scholar