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Goldie M-groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

K. C. Chowdhury
K. C. Chowdhury
Affiliation:
Gauhati UniversityGuwahati 781014 Assam, India
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Abstract

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If (G+) is a group and M is a nonempty set of endomorphisms of G operating on the left then G is said to be M-Goldie when (i) G has no infinite independent family of nonzero M-subgroups, and (ii) annihilators in M of subsets of G satisfy the a.c.c. (under set inclusion). Here we prove some results, analogous to those of a Noetherian module in some special cases, even when the set M of operators has no other algebraic structure than the existence of a zero element or in some cases M is at most a finite dimensional commutative near-ring. Precisely speaking, we prove that the collection of associated operating sets of G is finite and there exists a primary decomposition of 0 of such a Goldie M-group, and then if M is a finite dimensional commutative near-ring with unity, for any x belonging to each associated operating set of G, a power of it belongs to the annthilator of G.

MSC classification

Secondary: 20B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 237 - 246
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Barshay, J., Topics in ring theory, W. A. Benjamin, New York, 1969.Google Scholar
[2]Chatters, A. W. and Hazarnavis, C. R., Rings with chain conditions, Pitman Advanced Publishing Program, Boston, 1980.Google Scholar
[3]Chowdhury, K. C. and Masum, A., ‘A note on regular left Goldie near-rings’, Nat. Acad. Sci. Lett. (India) 12 (1989).Google Scholar
[4]Jategaonkar, A. V., Left principal ideal rings, Springer-Verlag Lecture Notes No. 123, 1970.CrossRefGoogle Scholar
[5]Oswald, A., ‘Near rings with chain conditions on right annihilators’, Proc. Edinburgh Math. Soc. 23 (1980), 123–127.CrossRefGoogle Scholar