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On a new variation of injective modules

Year 2019, , 702 - 711, 01.02.2019
https://doi.org/10.31801/cfsuasmas.464103

Abstract

In this paper, we provide various properties of GE and GEE-modules, a new variation of injective modules. We call M a GE-module if it has a g-supplement in every extension N and, we call also M a GEE-module if it has ample g-supplements in every extension N. In particular, we prove that every semisimple module is a GE-module. We show that a module M is a GEE-module if and only if every submodule is a GE-module. We study the structure of GE and GEE-modules over Dedekind domains. Over Dedekind domains the class of GE-modules lies between WS-coinjective modules and Zöschinger's modules with the property (E). We also prove that, if a ring R is a local Dedekind domain, an R-module M is a GE-module if and only if M≅(R^{∗})ⁿ⊕K⊕N, where R^{∗} is the completion of R, K is injective and N is a bounded module.

References

  • Alizade, R., Bilhan, G., Smith, P.F., Modules whose maximal submodules have supplements, Comm. in Algebra, 29(6), (2001), 2389-2405.
  • Alizade, R., Demirci, Y.M., Durgun, Y., Pusat, D., The proper class generated by weak supplements, Comm. in Algebra, 42, (2014),56-72.
  • Alizade R., Büyükaşık, E., Extensions of weakly supplemented modules, Math. Scand., 103, (2008), 161-168.
  • Byrd, K.A., Rings whose quasi-injective modules are semisimple, Proc. Amer. Math. Soc., 33(2), (1972), 235-240.
  • Clark, J., Lomp, C., Vanaja, N., Wisbauer, R., Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics-Birkhäuser-Basel, (2006), 406.
  • Koşar, B., Nebiyev, C., Sökmez, N., G-supplemented modules, Ukrainian Mathematical Journal, 67(6), (2015), 975-980.
  • Çalışıcı, H., Türkmen, E., Modules that have a supplement in every cofinite extension, Georgian Math. J., 19, (2012), 209-216.
  • Hausen, J., Supplemented modules over Dedekind domains, Pac. J. Math., 100(2), (1982), 387-402.
  • Özdemir, S., Rad-supplementing modules, J. Korean Math. Soc., 53(2), (2016), 403-414.
  • Sharpe, D.W., Vamos, P., Injective Modules, Cambridge University Press, (1972), 190.
  • Smith, P.F., Finitely generated supplemented modules are amply supplemented, The Arabian Journal for Science And Engineering, 25(2C), (2000), 69-79.
  • Türkmen, B.N., Modules that have a supplement in every coatomic extension, Miskolc Mathematical Notes, 16(1), (2015), 543-551.
  • Wisbauer, R., Foundations of Modules and Ring Theory, Gordon and Breach, (1991), 606.
  • Zhou, D.X., Zhang X.R., Small-essential submodules and morita duality, Southeast Asian Bulletin of Mathematics, 3, (2011), 1051-1062.
  • Zöschinger, H., Modules that have a supplement in every extension, Math. Scand., 32, (1974), 267-287.
Year 2019, , 702 - 711, 01.02.2019
https://doi.org/10.31801/cfsuasmas.464103

Abstract

References

  • Alizade, R., Bilhan, G., Smith, P.F., Modules whose maximal submodules have supplements, Comm. in Algebra, 29(6), (2001), 2389-2405.
  • Alizade, R., Demirci, Y.M., Durgun, Y., Pusat, D., The proper class generated by weak supplements, Comm. in Algebra, 42, (2014),56-72.
  • Alizade R., Büyükaşık, E., Extensions of weakly supplemented modules, Math. Scand., 103, (2008), 161-168.
  • Byrd, K.A., Rings whose quasi-injective modules are semisimple, Proc. Amer. Math. Soc., 33(2), (1972), 235-240.
  • Clark, J., Lomp, C., Vanaja, N., Wisbauer, R., Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics-Birkhäuser-Basel, (2006), 406.
  • Koşar, B., Nebiyev, C., Sökmez, N., G-supplemented modules, Ukrainian Mathematical Journal, 67(6), (2015), 975-980.
  • Çalışıcı, H., Türkmen, E., Modules that have a supplement in every cofinite extension, Georgian Math. J., 19, (2012), 209-216.
  • Hausen, J., Supplemented modules over Dedekind domains, Pac. J. Math., 100(2), (1982), 387-402.
  • Özdemir, S., Rad-supplementing modules, J. Korean Math. Soc., 53(2), (2016), 403-414.
  • Sharpe, D.W., Vamos, P., Injective Modules, Cambridge University Press, (1972), 190.
  • Smith, P.F., Finitely generated supplemented modules are amply supplemented, The Arabian Journal for Science And Engineering, 25(2C), (2000), 69-79.
  • Türkmen, B.N., Modules that have a supplement in every coatomic extension, Miskolc Mathematical Notes, 16(1), (2015), 543-551.
  • Wisbauer, R., Foundations of Modules and Ring Theory, Gordon and Breach, (1991), 606.
  • Zhou, D.X., Zhang X.R., Small-essential submodules and morita duality, Southeast Asian Bulletin of Mathematics, 3, (2011), 1051-1062.
  • Zöschinger, H., Modules that have a supplement in every extension, Math. Scand., 32, (1974), 267-287.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Review Articles
Authors

Ali Pancar This is me 0000-0003-3870-9210

Burcu Nişancı Türkmen 0000-0001-7900-0529

Celil Nebiyev 0000-0002-7992-7225

Ergül Türkmen This is me 0000-0002-7082-1176

Publication Date February 1, 2019
Submission Date November 22, 2017
Acceptance Date March 10, 2018
Published in Issue Year 2019

Cite

APA Pancar, A., Nişancı Türkmen, B., Nebiyev, C., Türkmen, E. (2019). On a new variation of injective modules. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 702-711. https://doi.org/10.31801/cfsuasmas.464103
AMA Pancar A, Nişancı Türkmen B, Nebiyev C, Türkmen E. On a new variation of injective modules. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):702-711. doi:10.31801/cfsuasmas.464103
Chicago Pancar, Ali, Burcu Nişancı Türkmen, Celil Nebiyev, and Ergül Türkmen. “On a New Variation of Injective Modules”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 702-11. https://doi.org/10.31801/cfsuasmas.464103.
EndNote Pancar A, Nişancı Türkmen B, Nebiyev C, Türkmen E (February 1, 2019) On a new variation of injective modules. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 702–711.
IEEE A. Pancar, B. Nişancı Türkmen, C. Nebiyev, and E. Türkmen, “On a new variation of injective modules”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 702–711, 2019, doi: 10.31801/cfsuasmas.464103.
ISNAD Pancar, Ali et al. “On a New Variation of Injective Modules”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 702-711. https://doi.org/10.31801/cfsuasmas.464103.
JAMA Pancar A, Nişancı Türkmen B, Nebiyev C, Türkmen E. On a new variation of injective modules. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:702–711.
MLA Pancar, Ali et al. “On a New Variation of Injective Modules”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 702-11, doi:10.31801/cfsuasmas.464103.
Vancouver Pancar A, Nişancı Türkmen B, Nebiyev C, Türkmen E. On a new variation of injective modules. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):702-11.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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