Maximal Non λ-Subrings

Abstract

Let R be a commutative ring with unity. The notion of maximal non λ-subrings is introduced and studied. A ring R is called a maximal non λ-subring of a ring T if RT is not a λ-extension, and for any ring S such that RST, ST is a λ-extension. We show that a maximal non λ-subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non λ-subring. If R is a maximal non λ-subring of a field K, where R is integrally closed in K, then K is the quotient field of R and R is a Prüfer domain. The equivalence of a maximal non λ-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non λ-subrings of a field.

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

References

  1. [1]

    A. Ayache, A. Jaballah: Residually algebraic pairs of rings. Math. Z. 225 (1997), 49–65.

    MathSciNet  Article  Google Scholar 

  2. [2]

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

    MathSciNet  MATH  Google Scholar 

  3. [3]

    A. Azarang, O. A. S. Karamzadeh: On the existence of maximal subrings in commutative Artinian rings. J. Algebra Appl. 9 (2010), 771–778.

    MathSciNet  Article  Google Scholar 

  4. [4]

    A. Azarang, G. Oman: Commutative rings with infinitely many maximal subrings. J. Algebra Appl. 13 (2014), Article ID 1450037, 29 pages.

    MathSciNet  Article  Google Scholar 

  5. [5]

    A. Badawi: On 2-absorbing ideals of commutative rings. Bull. Aust. Math. Soc. 75 (2007), 417–429.

    MathSciNet  Article  Google Scholar 

  6. [6]

    E. Bastida, R. Gilmer: Overrings and divisorial ideals of rings of the form D + M. Mich. Math. J. 20 (1973), 79–95.

    MathSciNet  Article  Google Scholar 

  7. [7]

    M. Ben Nasr, N. Jarboui: On maximal non-valuation subrings. Houston J. Math. 37 (2011), 47–59.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    D. E. Davis: Overrings of commutative rings III: Normal pairs. Trans. Am. Math. Soc. 182 (1973), 175–185.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    D. E. Dobbs: On INC-extensions and polynomials with unit content. Can. Math. Bull. 23 (1980), 37–42.

    MathSciNet  Article  Google Scholar 

  10. [10]

    D. E. Dobbs, G. Picavet, M. Picavet-L’Hermitte: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371 (2012), 391–429.

    MathSciNet  Article  Google Scholar 

  11. [11]

    D. E. Dobbs, J. Shapiro: Normal pairs with zero-divisors. J. Algebra Appl. 10 (2011), 335–356.

    MathSciNet  Article  Google Scholar 

  12. [12]

    E. G. Evans, Jr.: A generalization of Zariski’s main theorem. Proc. Am. Math. Soc. 26 (1970), 45–48.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    M. S. Gilbert: Extensions of Commutative Rings with Linearly Ordered Intermediate Rings. PhD Thesis. University of Tennessee, Knoxville, 1996. Available at https://search.proquest.com/docview/304271872?accountid=8179.

  14. [14]

    R. W. Gilmer, Jr.: Overrings of Prüfer domains. J. Algebra 4 (1966), 331–340.

    MathSciNet  Article  Google Scholar 

  15. [15]

    R. Gilmer: Some finiteness conditions on the set of overrings of an integral domain. Proc. Am. Math. Soc. 131 (2003), 2337–2346.

    MathSciNet  Article  Google Scholar 

  16. [16]

    R. W. Gilmer, Jr., J. F. Hoffmann: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60 (1975), 81–85.

    Article  Google Scholar 

  17. [17]

    A. Jaballah: Maximal non-Prüfer and maximal non-integrally closed subrings of a field. J. Algebra Appl. 11 (2012), Article ID 1250041, 18 pages.

    MathSciNet  Article  Google Scholar 

  18. [18]

    I. Kaplansky: Commutative Rings. University of Chicago Press, Chicago, 1974.

    Google Scholar 

  19. [19]

    R. Kumar, A. Gaur: On λ-extensions of commutative rings. J. Algebra Appl. 17 (2018), Article ID 1850063, 9 pages.

    MathSciNet  Article  Google Scholar 

  20. [20]

    I. J. Papick: Topologically defined classes of going-down domains. Trans. Am. Math. Soc. 219 (1976), 1–37.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgment

The authors thank the referee for his fruitful comments which helped to improve the quality of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Atul Gaur.

Additional information

The first author was supported by Junior Research Fellowship from UGC, India.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kumar, R., Gaur, A. Maximal Non λ-Subrings. Czech Math J 70, 323–337 (2020). https://doi.org/10.21136/CMJ.2019.0298-18

Download citation

Keywords

  • maximal non λ-subring
  • λ-extension
  • integrally closed extension
  • valuation domain

MSC 2010

  • 13B02
  • 13B22
  • 13A18