Maximal Non λ-Subrings


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.

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  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

  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 

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

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Correspondence to Atul Gaur.

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

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Kumar, R., Gaur, A. Maximal Non λ-Subrings. Czech Math J 70, 323–337 (2020).

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

MSC 2010

  • 13B02
  • 13B22
  • 13A18