On the Decomposition of Nonsingular CS-Modules

Clark, John ; Dung, Nguyen Viet

Canadian mathematical bulletin, Tome 39 (1996) no. 3, pp. 257-265

Résumé

It is shown that if M is a nonsingular CS-module with an indecomposable decomposition M = ⊕i∊I Mi , then the family {Mi | i € I} is locally semi-T"- nilpotent. This fact is used to prove that any nonsingular self-generator Σ-CS module is a direct sum of uniserial Noetherian quasi-injective submodules. As an application, we provide a new proof of Goodearl's characterization of non-singular rings over which all nonsingular right modules are projective.

Détail

DOI : 10.4153/CMB-1996-033-4

Mots-clés : Primary: 16D70, secondary: 16D40, 16D50, 16D80, nonsingular module, CS-module, indecomposable module, quasi-injective module, uniserial module, hereditary ring

Bibliographie

[AF] Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, Second edition, Graduate Texts in Math. 13, Springer-Verlag, New York, Heidelberg, Berlin, 1992. Google Scholar

[C] Cateforis, V. C., Two-sidedsemisimple maximal quotient rings, Trans. Amer. Math. Soc. 149(1970), 339– 349. Google Scholar

[CH] Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions, Research Notes in Math. Series 44, Pitman, London, 1980. Google Scholar

[CW] Clark, J. and Wisbauer, R., Σ-extending modules, J. Pure Appl. Algebra 104(1995), 19–32. Google Scholar

[D1] Dung, N. V., On indecomposable decompositions of CS-modules, J. Austral. Math. Soc. Ser. A, to appear. Google Scholar

Dung, N. V., On indecomposable decompositions of CS-modules II, J. Pure Appl. Algebra, to appear. Google Scholar

[DHSW] Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending Modules, Pitman Research Notes in Math. Series 313, Longman, Harlow, 1994. Google Scholar

[DS] Dung, N. V. and Smith, P. F., Σ-CS modules, Comm. Algebra 22(1994), 83–93. Google Scholar

Garcia, J. L. and Dung, N. V., Some decomposition properties of injective and pure-injective modules, Osaka J. Math. 31(1994), 95–108. Google Scholar

Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are n-injective, Comm. Algebra 6(1978), 59–73. Google Scholar

[Gl] Goodearl, K. R., Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124(1972). Google Scholar

[G2] Goodearl, K. R., Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, New York, Basel, 1976. Google Scholar

Kamal, M. A. and Muller, B. J., The structure of extending modules over Noetherian rings, Osaka J. Math. 25(1988), 539–551. Google Scholar

[MM] Mohamed, S. H. and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambridge Univ. Press, Cambridge, 1990. Google Scholar

Oshiro, K., Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13(1984), 310–338. Google Scholar

Osofsky, B. L., Rings all of whose finitely generated modules are injective, Pacific J. Math. 14(1964), 645–650. Google Scholar

[O2] Osofsky, B. L., Homological dimension and cardinality, Trans. Amer. Math. Soc. 151(1970), 641–649. Google Scholar

[S] Stenstrôm, B., Rings of Quotients, Springer-Verlag, Berlin, Heidelberg, New York, 1975. Google Scholar

Tachikawa, H., Quasi-Frobenius Rings and Generalizations. QF-3 and QF-1 Rings, Lecture Notes in Math. 351, Springer-Verlag, Berlin, New York, 1973. Google Scholar

Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991. Google Scholar

Zimmermann-Huisgen, B., Endomorphism rings of s elf-generators, Pacific J. Math. 61(1975), 587–602. Google Scholar