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PURE-INJECTIVITY FROM A DIFFERENT PERSPECTIVE

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  14 March 2017

S. R. LÓPEZ-PERMOUTH ,
J. MASTROMATTEO ,
Y. TOLOOEI  and
B. UNGOR
S. R. LÓPEZ-PERMOUTH
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA e-mails: lopez@ohio.edu, joseph.mastromatteo@gmail.com
J. MASTROMATTEO
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA e-mails: lopez@ohio.edu, joseph.mastromatteo@gmail.com
Y. TOLOOEI
Affiliation:
Department of Mathematics, Faculty of Science, Razi University, Kermanshah, 67149, Iran e-mail: y.toloei@razi.ac.ir
B. UNGOR
Affiliation:
Department of Mathematics, Ankara University, 06100 Ankara, Turkey e-mail: bungor@science.ankara.edu.tr
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Abstract

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The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

MSC classification

Primary: 16D10: General module theory
Secondary: 16D80: Other classes of modules and ideals
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 135 - 151
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Alahmadi, A. N., Alkan, M. and López-Permouth, S. R., Poor modules: The opposite of injectivity, Glasgow Math. J. 52 (A) (2010), 7–17.Google Scholar
2. Al Thani, N. M., Pure Baer injective modules, Internat. J. Math. Math. Sci. 20 (3) (1997), 529–538.Google Scholar
3. Aydogdu, P. and López-Permouth, S. R., An alternative perspective on injectivity of modules, J. Algebra 338 (1) (2011), 207–219.Google Scholar
4. Bican, L., El Bashir, R. and Enochs, E., All modules have flat covers, Bull. London Math. Soc. 33 (4) (2001), 385–390.CrossRefGoogle Scholar
5. Er, N., López-Permouth, S. R. and Sokmez, N., Rings whose modules have maximal and minimal injectivity domains, J. Algebra 330 (1) (2011), 404–417.Google Scholar
6. Harmanci, A., López-Permouth, S. R. and Ungor, B., On the pure-injectivity profile of a ring, Comm. Algebra 43 (11) (2015), 4984–5002.CrossRefGoogle Scholar
7. Lam, T. Y., Lectures on modules and rings (Springer-Verlag, New York, 1999).Google Scholar
8. López-Permouth, S. R. and Simental, J., Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra 362 (2012), 56–69.Google Scholar
9. Maddox, B. H., Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155–158.CrossRefGoogle Scholar
10. Prest, M., Pure-injective modules, Arab. J. Sci. Eng. Sect. A Sci. 34 (1D) (2009), 175–191.Google Scholar
11. Rotman, J. J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
12. Stenström, B., Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323–329.Google Scholar
13. Tuganbaev, A., Rings close to regular, Mathematics and its Applications, 545, Kluwer Academic Publishers, Dordrecht, 2002.Google Scholar
14. Warfield, R. B. Jr, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719.Google Scholar
15. Wisbauer, R., Foundations of module and ring theory (Gordon and Breach Science Publishers, Philadelphia, 1991).Google Scholar
16. Xu, J., Flat covers of modules (Springer-Verlag, Berlin, Heidelberg, 1996).Google Scholar