Abstract
ABSTRACT
We study the structure of semiperfect, CS-Modules with essential socle. We call the module M CSSES-module if M is semiperfect, CS-module with essential socle. We will call the ring R right CSSES-ring if the right R-module RR is CSSESmodule. In this note among others we prove that [i] If R is right CF and left GINring, then R is QF-ring if and only if R is right CS-ring if and only R is CSSESring. [ii] Every left Kasch right CF-ring is right CSSES-ring. [iii] If R is left Kasch and right IN-ring with equal left and right socles, then R is CSSES-ring.
Keywords
Key Words: CSSES-module and ring QF-ring Kasch ring CEP-ring semiperfect module and ring CF-ring
References
- Anderson F. W. and Fuller K. R., Rings and Categories of Modules, Springer-Verlag , New York, (1974).
- Camillo V., Nicholson W. K. and Yousif M. F., “Ikeda-Nakayama Rings”, J.Algebra, 226, 001-1010, (2000).
- Chen J., Ding N. and Yousif M. F., “On a Generalization of injective Rings”, Comm, Algebra , 31(10), 5105-5116, (2003).
- Chen J. and Ding N., “General Principally Injective Rings”, Comm. Algebra, 27(5), 2097-2116, (1999).
- Goodearl K. R., Ring Theory : Nonsingular Rings and Modules, Monographs on Pure and Applied Mathematics Vol. 33., Dekker, New York, (1976).
- Hajarnavis C. R. and Northon N. C., “On Dual Rings and Their Modules”, J.Algebra 93, 253-266, (1985).
- Ikeda M. and Nakayama T., “On Some Characteristic Properties of Quasi-Frobenius and Regular Rings”, Proc. Am. Math. Soc. 5, 15-19, (1954).
- Jain S. K. and Lopez-Permouth S. R., “Rings Whose Cyclics are Essentially Embeddable in Projectives”, J. Algebra. , 257-269, (1990).
- John B., “Annihilator Conditions in Noetherian Rings”, J. Algebra 49, 222-224, (1977).
- Mohamed S. H. and Müller B. J., Continuous and Discrete Modules, L.M.S. Lecture Notes Vol. 147. Cambridge University Press, Cambridge, UK, (1990).
- Nam S. B., Kim N. K. and Kim J. Y., “On Simple Singular GP-injective Modules”, Comm. Algebra, 27(5), 1683- , (1999).
- Nicholson W. K. and Yousif M. F., “On Perfect Simple-injective Rings”, Proc. Am. Math. Soc.,125, 979-985, (1997).
- Nicholson W. K. and Yousif M. F., Quasi-Frobenius Rings, Cambridge University Press, Cambridge Tracts in Mathematics. 158, (2003).
- Osofsky B., “a Generalization of Quasi-Frobenius Rings”, J. Algebra ,4, 373-387, (1966).
- Pardo J. L. G. and Asensio P. A. G., “Rings with Finite Essential Socle”, Proc. Am. Math. Soc. ,125, 971-977, (1997).
- Pardo J.L.G. and Asensio P.A.G., “Essential Embedding of Cyclic Modules in Projetives”, Trans. Am. Math. Soc. , , 4343-4353, (1997).
- Wisbauer R., Foundations of Module and Ring Theory, Gordon and Breach: Readiding, MA, (1991).
- Wisbauer R. ,Yousif M. F. and Zhou Y., “Ikeda-Nakayama Modules”, Beitraege zur Algebra und Geometrie , 43(1), 119, (2002).
- Yousif M. F and Zhou Y.,”Semiregular, Semiperfect and Perfect Rings Relative to an Ideal”, Rocky M. Jour. Math., (4), (2002).
- Zhou Y.,”Generalizations of Perfect, Semiperfect, and Semiregular Rings”, Algebra Colloquium, 7:3, 305-318, (2000).
Abstract
Bu çalışmada has desteğee sahip yarıtam CS-modüllerin yapısını araştıracağız. Eğer M modülü has desteğe sahip yarıtam CS-modülse M modülüne CSSES-modül denir. Sağ R-modül RR CSSES-modül ise R halkasına sağ CSSES-halkası denir. Bu çalışmada, diğer ispatladıklarımız yanında, aşağıdakilari de ispalayacağız: [i] Eğer R halkası sağ CF ise ve sol GIN-halka ise, o zaman R bir QF-halkadır ancak ve ancak R bir sağ CS-halkadır ancak ve ancak R bir sağ CSSES- halkadır. [ii] Her sol Kasch ve sağ CF-halka sağ CSSES-halkadır. [iii] Eğer R sağ ve sol destekleri eşit sol Kasch sağ IN-halka ise , o zaman R CSSES-halkadır
References
- Anderson F. W. and Fuller K. R., Rings and Categories of Modules, Springer-Verlag , New York, (1974).
- Camillo V., Nicholson W. K. and Yousif M. F., “Ikeda-Nakayama Rings”, J.Algebra, 226, 001-1010, (2000).
- Chen J., Ding N. and Yousif M. F., “On a Generalization of injective Rings”, Comm, Algebra , 31(10), 5105-5116, (2003).
- Chen J. and Ding N., “General Principally Injective Rings”, Comm. Algebra, 27(5), 2097-2116, (1999).
- Goodearl K. R., Ring Theory : Nonsingular Rings and Modules, Monographs on Pure and Applied Mathematics Vol. 33., Dekker, New York, (1976).
- Hajarnavis C. R. and Northon N. C., “On Dual Rings and Their Modules”, J.Algebra 93, 253-266, (1985).
- Ikeda M. and Nakayama T., “On Some Characteristic Properties of Quasi-Frobenius and Regular Rings”, Proc. Am. Math. Soc. 5, 15-19, (1954).
- Jain S. K. and Lopez-Permouth S. R., “Rings Whose Cyclics are Essentially Embeddable in Projectives”, J. Algebra. , 257-269, (1990).
- John B., “Annihilator Conditions in Noetherian Rings”, J. Algebra 49, 222-224, (1977).
- Mohamed S. H. and Müller B. J., Continuous and Discrete Modules, L.M.S. Lecture Notes Vol. 147. Cambridge University Press, Cambridge, UK, (1990).
- Nam S. B., Kim N. K. and Kim J. Y., “On Simple Singular GP-injective Modules”, Comm. Algebra, 27(5), 1683- , (1999).
- Nicholson W. K. and Yousif M. F., “On Perfect Simple-injective Rings”, Proc. Am. Math. Soc.,125, 979-985, (1997).
- Nicholson W. K. and Yousif M. F., Quasi-Frobenius Rings, Cambridge University Press, Cambridge Tracts in Mathematics. 158, (2003).
- Osofsky B., “a Generalization of Quasi-Frobenius Rings”, J. Algebra ,4, 373-387, (1966).
- Pardo J. L. G. and Asensio P. A. G., “Rings with Finite Essential Socle”, Proc. Am. Math. Soc. ,125, 971-977, (1997).
- Pardo J.L.G. and Asensio P.A.G., “Essential Embedding of Cyclic Modules in Projetives”, Trans. Am. Math. Soc. , , 4343-4353, (1997).
- Wisbauer R., Foundations of Module and Ring Theory, Gordon and Breach: Readiding, MA, (1991).
- Wisbauer R. ,Yousif M. F. and Zhou Y., “Ikeda-Nakayama Modules”, Beitraege zur Algebra und Geometrie , 43(1), 119, (2002).
- Yousif M. F and Zhou Y.,”Semiregular, Semiperfect and Perfect Rings Relative to an Ideal”, Rocky M. Jour. Math., (4), (2002).
- Zhou Y.,”Generalizations of Perfect, Semiperfect, and Semiregular Rings”, Algebra Colloquium, 7:3, 305-318, (2000).
Details
| Primary Language | English |
|---|---|
| Journal Section | Mathematics |
| Authors | |
| Publication Date | August 13, 2010 |
| Published in Issue | Year 2005 Volume: 18 Issue: 3 |