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RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

Published online by Cambridge University Press:  30 March 2012

PINAR AYDOĞDU ,
NOYAN ER  and
NİL ORHAN ERTAŞ
PINAR AYDOĞDU
Affiliation:
Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey e-mail: [email protected]
NOYAN ER
Affiliation:
Department of Mathematics, University of Rio Grande, Rio Grande, OH 45674, USA e-mail: [email protected]
NİL ORHAN ERTAŞ
Affiliation:
Department of Mathematics, Karabük University, 78050 Karabük, Turkey e-mail: [email protected]
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Abstract

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Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

Keywords

Primary 16D1016D7016P2016P40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 605 - 617
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Ahsan, J., Rings all of whose cyclic modules are quasi-injective, Proc. London Math. Soc., 27 (3) (1973), 425–439.Google Scholar
2.Ahsan, J., On rings with quasi-injective cyclic modules, Proc. Edinb. Math. Soc., II Ser. 19 (1974), 139–145.Google Scholar
3.Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules, Pitman Res. Notes Math. Ser. 313 (1994).Google Scholar
4.Er, N., Rings whose modules are direct sums of extending modules, Proc. Amer. Math. Soc. 137 (7) (2009), 2265–2271.CrossRefGoogle Scholar
5.Facchini, A., Module theory Birkhäuser, Vol. 167 (Basel, Boston, Berlin, 1998).Google Scholar
6.Faith, C., Algebra II, Ring Theory, Vol. 191 (Springer Verlag, Berlin, New York, 1976).Google Scholar
7.Fuller, K. R., Rings of left invariant module type, Comm. Algebra. 6 (2) (1978), 153–167.Google Scholar
8.Huynh, D. V., Jain, S. K. and López-Permouth, S. R., When is a simple ring Noetherian?, J. Algebra. 184 (1996), 786–794.Google Scholar
9.Jain, S. K., Singh, S. and Srivastava, A. K., On Σ - q rings, J. Pure Appl. Algebra. 213 (2009), 969–976.Google Scholar
10.Jain, S. K., Singh, S. and Symonds, R. G., Rings whose proper cyclic modules are quasi-injective, Pacific J. Math. 67 (1976), 461–472.Google Scholar
11.Johnson, R. E. and Wong, E. T., Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260–268.Google Scholar
12.Klatt, G. B. and Levy, L. S., Pre-self injective ringss, Trans. Amer. Math. Soc. 137 (1969), 407–419.Google Scholar
13.Koehler, A., Rings with quasi-injective cyclic modules, Quarterly J. Math. 25 (1974), 51–55.Google Scholar
14.Lam, T. Y., Lectures on modules and rings (Springer, Berlin, Heidelberg and New York, 1999).Google Scholar
15.Osofsky, B. and Smith, P. F., Cyclic modules whose quotients have all complement submodules direct summands, J. Algebra 139 (1991), 342–354.Google Scholar
16.Osofsky, B., Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650.Google Scholar
17.Osofsky, B., Noncyclic injective modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384.Google Scholar
18.Puninski, G., Serial Rings (Kluwer Pub, Netherlands, 2001).Google Scholar
19.Singh, S., Indecomposable modules over Artinian right serial rings in Advances in Ring Theory, (Jain, S. K. and Rizvi, S. T. Editors) (Birkhäuser Verlag, Boston, 1997), 295–304.Google Scholar