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Rings characterized by their right ideals or cyclic modules

  • Dinh Van Huynh (a1), Nguyen V. Dung (a2) and Patrick F. Smith
Extract

It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right R-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right R-module is injective. This leads to the second general question: given a ring R all of whose cyclic right R-modules have a certain property, what can one say about R itself?

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References
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1.Anderson, F. W. and Fuller, K. W., Rings and Categories of Modules (Springer-Verlag, 1974).
2.Van Huynh, Dinh, A note on Artinian rings, Arch. Math. (Basel), 33 (1979), 546553.
3.Van Huynh, Dinh, Some characterizations of hereditarily Artinian rings, Glasgow Math. J. 28 (1986), 2123.
4.Van Huynh, Dinh and Dung, Nguyen V., A characterization of Artinian rings, Glasgow Math. J. 30(1988), 6773.
5.Eisenbud, D. and Griffith, P., The structure of serial rings, Pacific J. Math. 36 (1971), 109121.
6.Faith, C., Algebra II: Ring Theory (Springer-Verlag, 1976).
7.Kertesz, A. and Widiger, A., Artinsche Ringe mil artinshem Radikal, J. Reine Angew, Math. 242 (1970), 815.
8.Osofsky, B. L., Non-injective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 13831384.
9.Widiger, A., Lattice of radicals for hereditarily Artinian rings, Math. Nachr. 84 (1978), 301309.
10.Widiger, A. and Wiegandt, R., Theory of radicals for hereditarily Artinian rings, Acta Sc. Math. Szeged 39 (1977), 303312.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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