Skip to main content
SpringerLink
Log in

FMR-rings in some distinguished constructions

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

A ring R is called an FMR-ring if it is finite modulo all its maximal ideals. In this paper, we investigate some properties of FMR-rings in various ring extensions. Mainly we are concerned with the charachterization of FMR-rings arising from bi-amalgamated algebras and trivial ring extensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, D.D., Winderes, M.: Idealization of a module. J. Comm. Algebra 1, 3–56 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Azarang, A.: On maximal subrings. Far East J. Math. Sci. 32(1), 107–118 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Azarang, A.: On the existence of maximal subrings in commutative Noetherian rings. J. Algebra Appl. 14(1), 1450073 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Butts, H.S., Wade, L.I.: Two criteria for Dedekind domains. Am. Math. Mon. 73, 14–21 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chew, K.L., Lawn, S.: Residually finite rings. Canad. J. Math. 22, 92–101 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. D’Anna, M., Finocchiaro, C.A., Fontana, M.: Amalgamated algebras along an ideal, in: Commutative Algebra and Applications, Walter de Gruyter, Berlin, (2009), 155–172

  7. D’Anna, M., Finocchiaro, C.A., Fontana, M.: Properties of chains of prime ideals in an amalgamated algebra along an ideal. J. Pure Appl. Algebra 214, 1633–1641 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. El Khalfi, A., Kim, H., Mahdou, N.: Amalgamation extension in commutative ring theory: a survey. Moroccan J. Algebra Geometry Appl. 1(1), 139–182 (2022)

    Google Scholar 

  9. Finocchiaro, C.A., Tartarone, F.: On a topological characterization of Prüfer \(v\)-multiplication domains among essential domains. J. Comm. Algebra 8(4), 513–536 (2016)

    MATH  Google Scholar 

  10. Fuchs, L., Salce, L.: Modules over non–Noetherian Domains, Mathematical Surveys and Monographs, Vol. 84, AMS, (2001)

  11. Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York (1972)

    MATH  Google Scholar 

  12. Gilmer, R.: Commutative Semigroup Rings. The Univ. Chicago, Press (1984)

    MATH  Google Scholar 

  13. Gilmer, R., Heinzer, W.: Some countability conditions on commutative ring extensions. Trans. Amer. Math. Soc. 264, 217–234 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilmer, R., Heinzer, W.: Products of commutative rings and zero-dimensionality. Trans. Amer. Math. Soc. 331, 662–680 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gilmer, R., Teply, M.L.: Idempotents of commutative semigroup rings. Houston J. Math. 3(3), 369–385 (1977)

    MathSciNet  MATH  Google Scholar 

  16. Hutchins, H.C.: Examples of Commutative Rings, Polygonal Publishing House, (1981)

  17. Izelgue, L., Ouzzaouit, O., Hilbert rings : G(oldman)-rings issued from amalgamated algebras, J. Algebra and App. 16 (11) (2018), https://doi.org/10.1142/S0219498818500238

  18. Kabbaj, S., Louartiti, K., Tamekkante, M.: Bi-amalgamated algebras along ideals. J. Commut. Algebra 9(1), 65–87 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kaplansky, I.: Commutative Rings. The university of Chicago Press, Chicago (1974)

    MATH  Google Scholar 

  20. Levitz, K., Mott, J.: Rings with finite norm property. Canad. J. Math. 24, 557–565 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Long, T.S.: Ring Extensions involving Amalgamated Duplications, Ph.D. Thesis, George Mason University, (2014)

  22. Orzech, M., Ribes, L.: Residual finiteness and the Hopf property in rings. J. Algebra 15, 81–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  23. Parker, T., Gilmer, R.: Nilpotent elements of commutative semigroup rings. Mich. Math. J. 22, 97–108 (1975)

    MathSciNet  MATH  Google Scholar 

  24. Tamekkante, M., Bouba, E.M.: Bi-amalgamation of small weak global dimension. Int. Electron. J. Algebra 21, 127–136 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are extremely grateful to the anonymous referees for their valuable comments and suggestions that greatly helped to improve this paper.

Author information

Authors and Affiliations

  1. The Higher School of Education and Training, Ibn Zohr University, Agadir, Morocco

    Omar Ouzzaouit

  2. Laboratory of Mathematics and Applications (LMA), Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

    Omar Ouzzaouit & Ali Tamoussit

  3. Department of Mathematics, The Regional Center for Education and Training Professions Souss-Massa, Inezgane, Morocco

    Ali Tamoussit

Authors
  1. Omar Ouzzaouit
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Tamoussit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omar Ouzzaouit.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ouzzaouit, O., Tamoussit, A. FMR-rings in some distinguished constructions. Rend. Circ. Mat. Palermo, II. Ser 72, 2311–2320 (2023). https://doi.org/10.1007/s12215-022-00797-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-022-00797-w

Keywords

Mathematics Subject Classification

Search

Navigation