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Rings in which idempotents generate maximal or minimal ideals

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Abstract

We characterize rings in which every left ideal generated by an idempotent different from 0 and 1 is either a maximal left ideal or a minimal left ideal. In the commutative case, we give a characterization in terms of topological properties of the maximal spectrum with the Zariski topology. We also consider a strictly weaker variant of this property, defined almost similarly, and characterize those rings that have the property in question.

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References

  1. Azarpanah, F.: Essential ideals in \(C(X)\). Period. Math. 31, 105–112 (1995)

    Article  MathSciNet  Google Scholar 

  2. Dube, T.: A note on the socle of certain types of \(f\)-rings. Bull. Iran. Math. Soc. 38, 517–528 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer, New York (1976)

    MATH  Google Scholar 

  4. Karamzadeh, O.A.S., Rostami, M.: On the intrinsic topology and some related ideals of \(C(X)\). Proc. Am. Math. Soc. 93(1), 179–184 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Lam, T.Y.: A First Course in Noncommutative Rings. Springer, New York (1991)

    Book  Google Scholar 

  6. Lambek, J.: Lectures on Rings and Modules. Chelsea Publ. Co., New York (1976)

    MATH  Google Scholar 

  7. Nicholson, W.K.: Lifting idempotents in exchange rings. Trans. Am. Math. Soc. 229, 229–278 (1977)

    Article  Google Scholar 

  8. Samei, K.: On the maximal spectrum of commutative semiprimitive rings. Coll. Math. 83(1), 5–13 (2000)

    Article  MathSciNet  Google Scholar 

  9. Steinberg, S.A.: Lattice-Ordered Rings and Modules. Springer, New York (2010)

    Book  Google Scholar 

  10. Taherifar, A.: Some new classes of topological spaces and annihilator ideals. Topol. Appl. 165, 84–97 (2014)

    Article  MathSciNet  Google Scholar 

  11. Varadarajan, K.: Clean, almost clean, potent commutative rings. J. Algebra Appl. 6, 671–685 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to the referee for suggestions (all of which have been incorporated) that helped improve both the presentation and content of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003, South Africa

    Themba Dube

  2. Department of Mathematics, Yasouj University, Yasouj, Iran

    Mojtaba Ghirati & Ali Taherifar

  3. Insa Centre Val de Loire, University of Orléans, Lifo ea 4022, Orléans, France

    Sajad Nazari

Authors
  1. Themba Dube
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  2. Mojtaba Ghirati
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  3. Sajad Nazari
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  4. Ali Taherifar
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Corresponding author

Correspondence to Themba Dube.

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Presented by W. Wm. McGovern

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The research of T. Dube was supported by the National Research Foundation of South Africa under Grant no. 113829.

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Dube, T., Ghirati, M., Nazari, S. et al. Rings in which idempotents generate maximal or minimal ideals. Algebra Univers. 81, 30 (2020). https://doi.org/10.1007/s00012-020-00660-y

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  • DOI: https://doi.org/10.1007/s00012-020-00660-y

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