Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T19:55:35.973Z Has data issue: false hasContentIssue false
  • English
  • Français

RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN*

Published online by Cambridge University Press:  24 June 2010

C. J. HOLSTON ,
S. K. JAIN  and
A. LEROY
C. J. HOLSTON
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA e-mail: [email protected]
S. K. JAIN
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA Department of Mathematics, King Abdulaziz University Jeddah, KSA e-mail: [email protected]
A. LEROY
Affiliation:
Department of Mathematics, University of Artois, Rue J. Souvraz, 62300 Lens, France e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

Keywords

16D5016D7016D80
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue A: Rings and Modules in Honour of Patrick F. Smith's 65th Birthday , July 2010 , pp. 103 - 110
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules (Longman Scientific and Technical, Harlow, UK, 1994).Google Scholar
2.Dung, N. V., Huynh, D. V. and Wisbauer, R., On modules with finite uniform and Krull dimension, Arch. Math. 57 (1991), 122–132.Google Scholar
3.Faith, C., Algebra: rings, modules and categories (Springer-Verlag, New York, 1973).Google Scholar
4.Faith, C., Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs 65 (AMS, Providence, RI, 1999).Google Scholar
5.Goel, S. C., Jain, S. K. and Singh, S., Rings whose cyclic modules are injective or projective, Proc. Amer. Math. Soc. 53 (1975), 16–18.Google Scholar
6.Huynh, D. V. and Rizvi, S. T., An affirmative answer to a question on noetherian rings, J. Algebra Appl. 7 (2008), 47–59.CrossRefGoogle Scholar
7.Huynh, D. V., Rizvi, S. T. and Yousif, M. F., Rings whose finitely generated modules are extending, J. Pure Appl. Algebra 111 (1996), 325–328.Google Scholar
8.Kaplansky, I., Topological representation of algebras II, Trans. Amer. Math. Soc. 68 (1950), 62–75.Google Scholar
9.Lam, T. Y., Lectures on modules and rings, Graduate Texts in Mathematics 189 (Springer-Verlag, New York, 1999).Google Scholar
10.Levy, L. S., Commutative rings whose homomorphic images are self- injective, Pacific J. Math. 18 (1966), 149–153.Google Scholar
11.Michler, G. O. and Villamayor, O. E., On rings whose simple modules are injective, J. Algebra 25 (1973), 185–201.CrossRefGoogle Scholar
12.Osofsky, B. and Smith, P. F., Cyclic modules whose quotients have all complement submodules direct summands, J. Algebra 139 (1991), 342–354.Google Scholar
13.Plubtieng, S. and Tansee, H., Conditions for a ring to be noetherian or artinian, Comm. Algebra 30 (2) (2002), 783–786.Google Scholar
14.Smith, P. F., Rings characterized by their cyclic modules, Can. J. Math. 24 (1979), 93–111.Google Scholar
15.Shock, R. C., Dual generalizations of the artinian and noetherian conditions, Pacific J. Math. 54 (1974), 227–235.Google Scholar
16.Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Reading, UK, 1991).Google Scholar