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Some characterizations of hereditarily artinian rings

Published online by Cambridge University Press:  18 May 2009

Dinh van Huynh
Dinh van Huynh
Affiliation:
Institute of Mathematics, P.O.Box 631 Bo hô Hanoi–Vietnam
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Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:

A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 21 - 23
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Chatters, A. W., A characterisation of right noetherian rings, Quart. J. Math. Oxford Ser. (2) 33 (1982), 65–69.CrossRefGoogle Scholar
2.Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions (Pitman, 1980).Google Scholar
3.van Huynh, Dinh, A note on artinian rings, Arch. Math. (Basel) 33 (1979), 546–553.CrossRefGoogle Scholar
4.van Huynh, Dinh, A note on rings with chain conditions, preprint (Institute of Mathematics, Hanoi, 1984).Google Scholar
5.Kertész, A. and Widiger, A., Artinsche Ringe mit artinschem Radikal, J. Reine Angew. Math. 242 (1970), 8–15.Google Scholar
6.Osofsky, B. L., Noncommutative rings whose cyclic modules have cyclic injective hulls, Pacific J. Math. 25 (1968), 331–340.CrossRefGoogle Scholar
7.Widiger, A., Zur Zerlegung artinscher Ringe, Publ. Math. Debrecen 21 (1974), 193–196.CrossRefGoogle Scholar
8.Widiger, A., Lattice of radicals for hereditarily artinian rings, Math. Nachr. 84 (1978), 301–309.CrossRefGoogle Scholar
9.Widiger, A. and Wiegandt, R., Theory of radicals for hereditarily artinian rings, Acta Sci. Math. (Szeged) 39 (1977), 303–312.Google Scholar