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CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES

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

Öz

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.

Kaynakça

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

Öz

Kaynakça

  • 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.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Farid Kourki Bu kişi benim

Rachid Tribak

Yayımlanma Tarihi 7 Ocak 2020
Yayımlandığı Sayı Yıl 2020

Kaynak Göster

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. Ocak 2020;27(27):178-193. doi:10.24330/ieja.663060
Chicago Kourki, Farid, ve Rachid Tribak. “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”. International Electronic Journal of Algebra 27, sy. 27 (Ocak 2020): 178-93. https://doi.org/10.24330/ieja.663060.
EndNote Kourki F, Tribak R (01 Ocak 2020) CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES. International Electronic Journal of Algebra 27 27 178–193.
IEEE F. Kourki ve R. Tribak, “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”, IEJA, c. 27, sy. 27, ss. 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 (Ocak 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 ve Rachid Tribak. “CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES”. International Electronic Journal of Algebra, c. 27, sy. 27, 2020, ss. 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.