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Serial Right Noetherian Rings

  • Surjeet Singh (a1)

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A module M is called a serial module if the family of its submodules is linearly ordered under inclusion. A ring R is said to be serial if RR as well as RR are finite direct sums of serial modules. Nakayama [8] started the study of artinian serial rings, and he called them generalized uniserial rings. Murase [5, 6, 7] proved a number of structure theorems on generalized uniserial rings, and he described most of them in terms of quasi-matrix rings over division rings. Warfield [12] studied serial both sided noetherian rings, and showed that any such indecomposable ring is either artinian or prime. He further showed that a both sided noetherian prime serial ring is an (R:J)-block upper triangular matrix ring, where R is a discrete valuation ring with Jacobson radical J. In this paper we determine the structure of serial right noetherian rings (Theorem 2.11).

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References

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1. Eisenbud, D. and Robson, J. C., Hereditary noethehan prime rings, J. Algebra 16 (1970), 86104.
2. Jategaonka, A. V., Left principal ideal rings, Springer-Verlag 123 (1970).
3. Johnson, R. E. and Wong, E. T., Quasi-infective modules and irreducible rings, J. Lon. Math Soc. 36 (1961), 260268.
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5. Murase, I., On the structure of generalized uniserial rings I, Sci. Papers, College Gen. Ed., Univ. Tokyo 13 (1963), 122.
6. Murase, I., On the structure oj generalized uniserial rings II, Sei. Papers, College Gen. Ed., Univ. Tokyo 13 (1963), 131158.
7. Murase, I., On the structure of generalized uniserial rings III, Sci. Papers College Gen. Ed., Univ. Tokyo 14 (1964), 1125.
8. Nakayama, T., On Froheniusean algebras 11, Ann. Math. 42 (1941), 121.
9. Singh, S., Quasi-injective and quasi-projective modules over hereditary noetherian prime rings, Can. J. Math. 26 (1974), 11731185.
10. Singh, S., Modules over hereditary noetherian prime rings 11, Can. J. Math. 28 (1976), 7382.
11. Singh, S., Some decomposition theorems in abelian groups and their generalizations, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 25 (1976), 183189.
12. Warfield, R. B. Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), 187222.
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Serial Right Noetherian Rings

  • Surjeet Singh (a1)

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