Research Article
BibTex RIS Cite

CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES

Year 2020, , 178 - 193, 07.01.2020
https://doi.org/10.24330/ieja.663060

Abstract

We introduce the notion of locally supplemented modules (i.e., modules for which every finitely generated submodule is supplemented). We show that a module $M$ is locally supplemented if and only if $M$ is a sum of local submodules. We characterize several classes of rings in terms of locally supplemented modules. Among others, we prove that a ring $R$ is a Camillo ring if and only if every finitely embedded $R$-module is locally supplemented. It is also shown that a ring $R$ is a Gelfand ring if and only if every $R$-module having a finite Goldie dimension is locally supplemented.

References

  • D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30(7) (2002), 3327-3336.
  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • I. Beck, $\Sigma$-injective modules, J. Algebra, 21(2) (1972), 232-249.
  • N. Bourbaki, Elements de Mathematique, Algebre Commutative, Chapitres 1 et 2, Masson, Paris, 1985.
  • E. Buyukasik and C. Lomp, Rings whose modules are weakly supplemented are perfect. Applications to certain ring extensions, Math. Scand., 105(1) (2009), 25-30.
  • V. Camillo, Homological independence of injective hulls of simple modules over commutative rings, Comm. Algebra, 6(14) (1978), 1459-1469.
  • J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006.
  • F. Couchot, Indecomposable modules and Gelfand rings, Comm. Algebra, 35(1) (2007), 231-241.
  • A. Facchini, Module Theory, Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Mathematics, 167, Birkhauser Verlag, Basel, 1998.
  • C. Faith, Minimal cogenerators over Osofsky and Camillo rings, Advances in Ring Theory (Granville, OH, 1996), Trends Math., Birkhauser Boston, Boston, MA, (1997), 105-118.
  • A. I. Generalov, $\omega$-cohigh purity in a category of modules, Mat. Zametki, 33(5) (1983), 402-408; translation from Mat. Zametki, 33(5) (1983), 785-796.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
  • S. McAdam, Deep decompositions of modules, Comm. Algebra, 26(12) (1998), 3953-3967.
  • S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990.
  • D. W. Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, No. 62, Cambridge University Press, London-New York, 1972.
  • R. C. Shock, Dual generalizations of the artinian and noetherian conditions, Paci c J. Math., 54(2) (1974), 227-235.
  • T. S. Shores, Decompositions of finitely generated modules, Proc. Amer. Math. Soc., 30(3) (1971), 445-450.
  • P. Vamos, The dual of the notion of "finitely generated", J. London Math. Soc., 43(1) (1968), 643-646.
  • P. Vamos, Classical rings, J. Algebra, 34(1) (1975), 114-129.
  • H. Zoschinger, Komplementierte Moduln uber Dedekindringen, J. Algebra, 29(1) (1974), 42-56.
  • H. Zoschinger, Gelfandringe und koabgeschlossene Untermoduln, Bayer. Akad. Wiss. Math.-Natur. KI. Sitzungsber., 3 (1982), 43-70.
Year 2020, , 178 - 193, 07.01.2020
https://doi.org/10.24330/ieja.663060

Abstract

References

  • D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30(7) (2002), 3327-3336.
  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • I. Beck, $\Sigma$-injective modules, J. Algebra, 21(2) (1972), 232-249.
  • N. Bourbaki, Elements de Mathematique, Algebre Commutative, Chapitres 1 et 2, Masson, Paris, 1985.
  • E. Buyukasik and C. Lomp, Rings whose modules are weakly supplemented are perfect. Applications to certain ring extensions, Math. Scand., 105(1) (2009), 25-30.
  • V. Camillo, Homological independence of injective hulls of simple modules over commutative rings, Comm. Algebra, 6(14) (1978), 1459-1469.
  • J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006.
  • F. Couchot, Indecomposable modules and Gelfand rings, Comm. Algebra, 35(1) (2007), 231-241.
  • A. Facchini, Module Theory, Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Mathematics, 167, Birkhauser Verlag, Basel, 1998.
  • C. Faith, Minimal cogenerators over Osofsky and Camillo rings, Advances in Ring Theory (Granville, OH, 1996), Trends Math., Birkhauser Boston, Boston, MA, (1997), 105-118.
  • A. I. Generalov, $\omega$-cohigh purity in a category of modules, Mat. Zametki, 33(5) (1983), 402-408; translation from Mat. Zametki, 33(5) (1983), 785-796.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
  • S. McAdam, Deep decompositions of modules, Comm. Algebra, 26(12) (1998), 3953-3967.
  • S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990.
  • D. W. Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, No. 62, Cambridge University Press, London-New York, 1972.
  • R. C. Shock, Dual generalizations of the artinian and noetherian conditions, Paci c J. Math., 54(2) (1974), 227-235.
  • T. S. Shores, Decompositions of finitely generated modules, Proc. Amer. Math. Soc., 30(3) (1971), 445-450.
  • P. Vamos, The dual of the notion of "finitely generated", J. London Math. Soc., 43(1) (1968), 643-646.
  • P. Vamos, Classical rings, J. Algebra, 34(1) (1975), 114-129.
  • H. Zoschinger, Komplementierte Moduln uber Dedekindringen, J. Algebra, 29(1) (1974), 42-56.
  • H. Zoschinger, Gelfandringe und koabgeschlossene Untermoduln, Bayer. Akad. Wiss. Math.-Natur. KI. Sitzungsber., 3 (1982), 43-70.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Farid Kourki This is me

Rachid Tribak

Publication Date January 7, 2020
Published in Issue Year 2020

Cite

APA Kourki, F., & Tribak, R. (2020). CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. International Electronic Journal of Algebra, 27(27), 178-193. https://doi.org/10.24330/ieja.663060
AMA Kourki F, Tribak R. CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. IEJA. January 2020;27(27):178-193. doi:10.24330/ieja.663060
Chicago Kourki, Farid, and Rachid Tribak. “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 178-93. https://doi.org/10.24330/ieja.663060.
EndNote Kourki F, Tribak R (January 1, 2020) CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. International Electronic Journal of Algebra 27 27 178–193.
IEEE F. Kourki and R. Tribak, “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”, IEJA, vol. 27, no. 27, pp. 178–193, 2020, doi: 10.24330/ieja.663060.
ISNAD Kourki, Farid - Tribak, Rachid. “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”. International Electronic Journal of Algebra 27/27 (January 2020), 178-193. https://doi.org/10.24330/ieja.663060.
JAMA Kourki F, Tribak R. CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. IEJA. 2020;27:178–193.
MLA Kourki, Farid and Rachid Tribak. “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 178-93, doi:10.24330/ieja.663060.
Vancouver Kourki F, Tribak R. CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. IEJA. 2020;27(27):178-93.